Triangle graph

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, the method of steepest descent or stationary phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.

The integral to be estimated is often of the form

$\underset{C}{\int }f(z)e^{\lambda g(z)}dz$

where C is a contour and λ is large. One version of the method of steepest descent deforms the contour of integration so that it passes through a zero of the derivative g′(z) in such a way that on the contour g is (approximately) real and has a maximum at the zero.

The method of steepest descent was first published by Template:Harvtxt, who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note Template:Harvtxt about hypergeometric functions. The contour of steepest descent has a minimax property, see Template:Harvtxt. Template:Harvtxt described some other unpublished notes of Riemann, where he used this method to derive the Riemann-Siegel formula.

A simple estimate

Let $f,S:{\mathbb{ℂ}}^n\to \mathbb{ℂ}$ and $C\subset {\mathbb{ℂ}}^n.$ If $M=\sup {\mathrm{\backslash limits}}_{x\in C}\mathrm{\Re }[S(x)]<\mathrm{\infty }$, where $\mathrm{\Re }[S(x)]$ denotes the real part, and there exists ${\lambda }_0\in \mathbb{ℝ},{\lambda }_0>0,$ such that

$\underset{C}{\int }|f(x)e^{{\lambda }_{0}S(x)}|dx<\mathrm{\infty },$

then the following estimate holds:

$|\underset{C}{\int }f(x)e^{\lambda S(x)}dx|⩽\text{const}\cdot e^{\lambda M},\mathrm{\forall }\lambda \in \mathbb{ℝ},\lambda ⩾{\lambda }_0.$

Proof of the simple estimate

Taking into account that $\left|e^{(\lambda -{\lambda }_0)(S(x)-M)}\right|⩽1$, we get

$\begin{array}{rl}|\underset{C}{\int }f(x)e^{\lambda S(x)}dx|& ⩽\underset{C}{\int }|f(x)|\left|e^{\lambda S(x)}dx\right|\equiv \underset{C}{\int }|f(x)|e^{\lambda M}\left|e^{{\lambda }_0[S(x)-M]}{e}^{(\lambda -{\lambda }_0)(S(x)-M)}dx\right|\\ & ⩽\underset{C}{\int }|f(x)|e^{\lambda M}\left|e^{{\lambda }_0[S(x)-M]}dx\right|=\underset{\text{const}}{\underbrace{e^{-{\lambda }_{0}M}\underset{C}{\int }|f(x)e^{{\lambda }_{0}S(x)}dx|}}\cdot e^{\lambda M}.\end{array}$

The case of a single non-degenerate saddle point

Basic notions and notation

Let $x$ be a complex n-dimensional vector, and

$S^{\prime }{\text{'}}_{xx}(x)\equiv \left(\frac{{\partial }^{2}S(x)}{\partial x_i\partial x_j}\right),1⩽i,j⩽n,$

denote the Hessian matrix for a function $S(x)$. If $\mathit{\phi }(x)=({\phi }_1(x),{\phi }_2(x),\dots ,{\phi }_k(x))$ is a vector function, then its Jacobian matrix is defined as

${{\mathit{\phi }}_x}^{\prime }(x)\equiv \left(\frac{\partial {\phi }_i(x)}{\partial x_j}\right),1⩽i⩽k,1⩽j⩽n.$

A non-degenerate saddle point, $z^0\in {\mathbb{ℂ}}^n$, of a holomorphic function $S(z)$ is a point where the function reaches an extremum (i.e., $\mathrm{\nabla }S(z^0)=0$) and has a non-vanishing determinant of the Hessian (i.e., $\det S^{\prime }{\text{'}}_{zz}(z^0)\ne 0$).

The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point:

Complex Morse Lemma

The Morse lemma for real-valued functions generalizes as follows for holomorphic functions: near a non-degenerate saddle point $z^0$ of a holomorphic function $S(z)$, there exist coordinates in terms of which $S(z)-S(z^0)$ is quadratic. Let $S$ be a holomorphic function with domain $W\subset {\mathbb{ℂ}}^n$, and let $z^0\in W$ be a non-degenerate saddle point of $S$, that is, $\mathrm{\nabla }S(z^0)=0$ and $\det S^{\prime }{\text{'}}_{zz}(z^0)\ne 0$. Then there exist neighborhoods $U\subset W$ of $z^0$ and $V\subset {\mathbb{ℂ}}^n$ of $w=0$, and a bijective holomorphic function $\mathit{\phi }:V\to U$ with $\mathit{\phi }(0)=z^0$ such that

$S(\mathit{\phi }(w))=S(z^0)+\frac{1}{2}\sum _{j=1}^n{\mu }_j{w}_j^2,\det {{\mathit{\phi }}_w}^{\prime }(0)=1,$

at all points $w\in V$. Here, the ${\mu }_j$ are the eigenvalues of the matrix ${S_{z^{″}z}}^{″}(z^0)$.

An illustration of Complex Morse Lemma

Proof of Complex Morse Lemma

The following proof is a straightforward generalization of the proof of the real Morse Lemma, which can be found in. We begin by demonstrating

Auxiliary Statement. Let $f:{\mathbb{ℂ}}^n\to \mathbb{ℂ}$ be holomorphic in a neighborhood of the origin and $f(0)=0$. Then in some neighborhood, there exist functions $g_i:{\mathbb{ℂ}}^n\to \mathbb{ℂ}$ such that $f(z)=\sum _{i=1}^n{z}_i{g}_i(z)$, where each $g_i$ is holomorphic and $g_i(0)={\frac{\partial f(z)}{\partial z_i}|}_{z=0}$.

Proof of Auxiliary Statement

From the identity

$f(z)=\underset{0}{\overset{1}{\int }}\frac{d}{dt}f(t{z}_1,\dots ,t{z}_n)dt=\sum _{i=1}^n{z}_i\underset{0}{\overset{1}{\int }}{\frac{\partial f(z)}{\partial z_i}|}_{z=(t{z}_1,\dots ,t{z}_n)}dt,$

we conclude that $g_i(z)=\underset{0}{\overset{1}{\int }}{\frac{\partial f(z)}{\partial z_i}|}_{z=(t{z}_1,\dots ,t{z}_n)}dt$ and $g_i(0)={\frac{\partial f(z)}{\partial z_i}|}_{z=0}$.

Without loss of generality, we translate the origin to $z^0$, such that $z^0=0$ and $S(0)=0$. Using the Auxiliary Statement, we have

$S(z)=\sum _{i=1}^n{z}_i{g}_i(z).$

Since the origin is a saddle point,

${\frac{\partial S(z)}{\partial z_i}|}_{z=0}=g_i(0)=0,$

we can also apply the Auxiliary Statement to the functions $g_i(z)$ and obtain

$S(z)=\sum _{i,j=1}^n{z}_i{z}_j{h}_{ij}(z).$

(1)

Recall that an arbitrary matrix $A$ can be represented as a sum of symmetric $A^{(s)}$ and anti-symmetric $A^{(a)}$ matrices,

$A_{ij}=A_{ij}^{(s)}+A_{ij}^{(a)},A_{ij}^{(s)}=(A_{ij}+A_{ji})/2,A_{ij}^{(a)}=(A_{ij}-A_{ji})/2.$

The contraction of any symmetric matrix $B$ with an arbitrary matrix $A$ is

$\sum _{i,j}{B}_{ij}{A}_{ij}=\sum _{i,j}{B}_{ij}{A}_{ij}^{(s)},$

(2)

i.e., the anti-symmetric component of $A$ does not contribute because

$\sum _{i,j}{B}_{ij}{C}_{ij}=\sum _{i,j}{B}_{ji}{C}_{ji}=-\sum _{i,j}{B}_{ij}{C}_{ij}=0.$

Thus, $h_{ij}(z)$ in equation (1) can be assumed to be symmetric with respect to the interchange of the indices i and j. Note that

${\frac{{\partial }^{2}S(z)}{\partial z_i\partial z_j}|}_{z=0}=2h_{ij}(0);$

hence, $\det (h_{ij}(0))\ne 0$ because the origin is a non-degenerate saddle point.

Let us show by induction that there are local coordinates $u=(u_1,\dots ,u_n)$, $z=\mathit{\psi }(u)$, $0=\mathit{\psi }(0)$, such that

$S(\mathit{\psi }(u))=\sum _{i=1}^n{u}_i^2.$

(3)

First, assume that there exist local coordinates $y=(y_1,\dots ,y_n)$, $z=\mathit{\varphi }(y)$, $0=\mathit{\varphi }(0)$, such that

$S(\mathit{\varphi }(y))=y_1^2+\cdots +y_{r-1}^2+\sum _{i,j=r}^n{y}_i{y}_j{H}_{ij}(y),$

(4)

where $H_{ij}$ is symmetric due to equation (2). By a linear change of the variables $(y_r,\dots ,y_n)$, we can assure that $H_{rr}(0)\ne 0$. From the chain rule, we have

$\frac{{\partial }^{2}S(\mathit{\varphi }(y))}{\partial y_i\partial y_j}=\sum _{l,k=1}^n{\frac{{\partial }^{2}S(z)}{\partial z_k\partial z_l}|}_{z=\mathit{\varphi }(y)}\frac{\partial {\varphi }_k}{\partial y_i}\frac{\partial {\varphi }_l}{\partial y_j}+\sum _{k=1}^n{\frac{\partial S(z)}{\partial z_k}|}_{z=\mathit{\varphi }(y)}\frac{{\partial }^2{\varphi }_k}{\partial y_i\partial y_j}$

$⟹S^{\prime }{\text{'}}_{yy}(\mathit{\varphi }(0))=\mathit{\varphi }{\text{'}}_y(0{)}^T{S}^{\prime }{\text{'}}_{zz}(0)\mathit{\varphi }{\text{'}}_y(0),\det \mathit{\varphi }{\text{'}}_y(0)\ne 0;$

whence,

$0\ne \det S^{\prime }{\text{'}}_{yy}(\mathit{\varphi }(0))=2^{r-1}\det (2H_{ij}(0)).$

The matrix $(H_{ij}(0))$ can be recast in the Jordan normal form: $(H_{ij}(0))=LJ{L}^{-1}$, were $L$ gives the desired non-singular linear transformation and the diagonal of $J$ contains non-zero eigenvalues of $(H_{ij}(0))$. If $H_{ij}(0)\ne 0$ then, due to continuity of $H_{ij}(y)$, it must be also non-vanishing in some neighborhood of the origin. Having introduced ${\tilde{H}}_{ij}(y)=H_{ij}(y)/H_{rr}(y)$, we write

$\begin{array}{rl}S(\mathit{\varphi }(y))=& y_1^2+\cdots +y_{r-1}^2+H_{rr}(y)\sum _{i,j=r}^n{y}_i{y}_j{\tilde{H}}_{ij}(y)\\ =& y_1^2+\cdots +y_{r-1}^2+H_{rr}(y)[y_r^2+2y_r\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y)+\sum _{i,j=r+1}^n{y}_i{y}_j{\tilde{H}}_{ij}(y)]\\ =& y_1^2+\cdots +y_{r-1}^2+H_{rr}(y)[{(y_r+\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y))}^2-{(\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y))}^2]\\ & +H_{rr}(y)\sum _{i,j=r+1}^n{y}_i{y}_j{\tilde{H}}_{ij}(y).\end{array}$

Motivated by the last expression, we introduce new coordinates $z=\mathit{\eta }(x)$, $0=\mathit{\eta }(0)$,

$x_r=\sqrt{H_{rr}(y)}(y_r+\sum _{j=r+1}^n{y}_j{\tilde{H}}_{rj}(y)),x_j=y_j,\mathrm{\forall }j\ne r.$

The change of the variables $y\leftrightarrow x$ is locally invertible since the corresponding Jacobian is non-zero,

${\frac{\partial x_r}{\partial y_k}|}_{y=0}=\sqrt{H_{rr}(0)}[{\delta }_{r,k}+\sum _{j=r+1}^n{\delta }_{j,k}{\tilde{H}}_{jr}(0)].$

Therefore,

$S(\mathit{\eta }(x))=x_1^2+\cdots +x_r^2+\sum _{i,j=r+1}^n{x}_i{x}_j{W}_{ij}(x).$

(5)

Comparing equations (4) and (5), we conclude that equation (3) is verified. Denoting the eigenvalues of $S^{\prime }{\text{'}}_{zz}(0)$ by ${\mu }_j$, equation (3) can be rewritten as

$S(\mathit{\phi }(w))=\frac{1}{2}\sum _{j=1}^n{\mu }_j{w}_j^2.$

(6)

Therefore,

$S^{\prime }{\text{'}}_{ww}(\mathit{\phi }(0))=\mathit{\phi }{\text{'}}_w(0{)}^T{S}^{\prime }{\text{'}}_{zz}(0)\mathit{\phi }{\text{'}}_w(0),$

(7)

From equation (6), it follows that $\det S^{\prime }{\text{'}}_{ww}(\mathit{\phi }(0))={\mu }_1\cdots {\mu }_n$. The Jordan normal form of $S^{\prime }{\text{'}}_{zz}(0)$ reads $S^{\prime }{\text{'}}_{zz}(0)=P{J}_z{P}^{-1}$, where $J_z$ is an upper diagonal matrix containing the eigenvalues and $\det P\ne 0$; hence, $\det S^{\prime }{\text{'}}_{zz}(0)={\mu }_1\cdots {\mu }_n$. We obtain from equation (7)

$\det S^{\prime }{\text{'}}_{ww}(\mathit{\phi }(0))={[\det \mathit{\phi }{\text{'}}_w(0)]}^2\det S^{\prime }{\text{'}}_{zz}(0)⟹\det \mathit{\phi }{\text{'}}_w(0)=\pm 1.$

If $\det \mathit{\phi }{\text{'}}_w(0)=-1$, then interchanging two variables assures that $\det \mathit{\phi }{\text{'}}_w(0)=+1$.

The asymptotic expansion in the case of a single non-degenerate saddle point

Assume

1. $f(z)$ and $S(z)$ are holomorphic functions in an open, bounded, and simply connected set ${\mathrm{\Omega }}_x\subset {\mathbb{ℂ}}^n$ such that the set $I_x={\mathrm{\Omega }}_x\cap {\mathbb{ℝ}}^n$ is connected;
2. $\mathrm{\Re }[S(z)]$ has a single maximum: $\max {\mathrm{\backslash limits}}_{z\in I_x}\mathrm{\Re }[S(z)]=\mathrm{\Re }[S(x^0)]$ for exactly one point $x^0\in I_x$;
3. $x^0$ is a non-degenerate saddle point (i.e., $\mathrm{\nabla }S(x^0)=0$ and $\det S^{\prime }{\text{'}}_{xx}(x^0)\ne 0$).

Then, the following asymptotic holds

$I(\lambda )\equiv \underset{I_x}{\int }f(x)e^{\lambda S(x)}dx={\left(\frac{2\pi }{\lambda }\right)}^{n/2}{e}^{\lambda S(x^0)}\prod _{j=1}^n(-{\mu }_j{)}^{-1/2}[f(x^0)+O\left({\lambda }^{-1}\right)],\lambda \to +\mathrm{\infty },$

(8)

where ${\mu }_j$ are eigenvalues of the Hessian $S^{\prime }{\text{'}}_{xx}(x^0)$ and $(-{\mu }_j{)}^{-1/2}$ are defined with arguments

$|\arg \sqrt{-{\mu }_j}|<\pi /4.$

(9)

This statement is a special case of more general results presented in.

Derivation of equation (8)

An illustration to the derivation of equation (8)

First, we deform the contour $I_x$ into a new contour $I{\text{'}}_x\subset {\mathrm{\Omega }}_x$ passing through the saddle point $x^0$ and sharing the boundary with $I_x$. This deformation does not change the value of the integral $I(\lambda )$. We employ the Complex Morse Lemma to change the variables of integration. According to the lemma, the function $\mathit{\phi }(w)$ maps a neighborhood $U\subset {\mathrm{\Omega }}_x$ ($x^0\in U$) onto a neighborhood ${\mathrm{\Omega }}_w$ containing the origin. The integral $I(\lambda )$ can be split into two: $I(\lambda )=I_0(\lambda )+I_1(\lambda )$, where $I_0(\lambda )$ is the integral over $U\cap I{\text{'}}_x$, while $I_1(\lambda )$ is over $I{\text{'}}_x\setminus (U\cap I{\text{'}}_x)$ (i.e., the remaining part of the contour $I{\text{'}}_x$). Since the latter region does not contain the saddle point $x^0$, the value of $I_1(\lambda )$ is exponentially smaller than $I_0(\lambda )$ as $\lambda \to +\mathrm{\infty }$; thus, $I_1(\lambda )$ is ignored. Introducing the contour $I_w$ such that $U\cap I{\text{'}}_x=\mathit{\phi }(I_w)$, we have

$I_0(\lambda )=\exp [\lambda S(x^0)]\underset{I_w}{\int }f[\mathit{\phi }(w)]\exp \left(\lambda \sum _{j=1}^n\frac{{\mu }_j}{2}{w}_j^2\right)|\det {{\mathit{\phi }}_w}^{\prime }(w)|dw.$

(10)

Recalling that $x^0=\mathit{\phi }(0)$ as well as $\det {{\mathit{\phi }}_w}^{\prime }(0)=1$, we expand the pre-exponential function into a Taylor series and keep just the leading zero-order term

$I_0(\lambda )\approx f(x^0)\exp [\lambda S(x^0)]\underset{{\mathbb{ℝ}}^n}{\int }\exp \left(\lambda \sum _{j=1}^n\frac{{\mu }_j}{2}{w}_j^2\right)dw=f(x^0)e^{\lambda S(x^0)}\prod _{j=1}^n\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{\lambda {\mu }_j{y}^2/2}dy.$

(11)

Here, we have substituted the integration region $I_w$ by ${\mathbb{ℝ}}^n$ because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term. The integrals in the r.h.s. of equation (11) can be expressed as

${\mathcal{ℐ}}_j=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{\lambda {\mu }_j{y}^2/2}dy=2\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda {\left(\sqrt{-{\mu }_j}y\right)}^2/2}dy=2\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\lambda |\sqrt{-{\mu }_j}{|}^2{y}^2\exp (2i\arg \sqrt{-{\mu }_j})/2}dy.$

(12)

From this representation, we conclude that condition (9) must be satisfied in order for the r.h.s. and l.h.s. of equation (12) to coincide. According to assumption 2, $\mathrm{\Re }[{S_{x^{″}x}}^{″}(x^0)]$ is a negatively defined quadratic form (viz., $\mathrm{\Re }({\mu }_j)<0$) implying the existence of the integral ${\mathcal{ℐ}}_j$, which is readily calculated

${\mathcal{ℐ}}_j=\frac{2}{\sqrt{-{\mu }_j}\sqrt{\lambda }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-{\xi }^2/2}d\xi =\sqrt{\frac{2\pi }{\lambda }}(-{\mu }_j{)}^{-1/2}.$

Equation (8) can also be written as

$I(\lambda )={\left(\frac{2\pi }{\lambda }\right)}^{n/2}{e}^{\lambda S(x^0)}{[\det (-{S_{x^{″}x}}^{″}(x^0))]}^{-1/2}[f(x^0)+O\left({\lambda }^{-1}\right)],$

(13)

where the branch of $\sqrt{\det (-{S_{x^{″}x}}^{″}(x^0))}$ is selected as follows

${[\det (-{S_{x^{″}x}}^{″}(x^0))]}^{-1/2}=\prod _{j=1}^n{\left|{\mu }_j\right|}^{-1/2}\exp [-i\mathrm{I}\mathrm{n}\mathrm{d}(-{S_{x^{″}x}}^{″}(x^0))],$

$\mathrm{I}\mathrm{n}\mathrm{d}(-{S_{x^{″}x}}^{″}(x^0))=\frac{1}{2}\sum _{j=1}^n\arg (-{\mu }_j),|\arg (-{\mu }_j)|<\pi /2.$

Consider important special cases:

• If $S(x)$ is real valued for real x and $x^0\in {\mathbb{ℝ}}^n$ (aka, the multidimensional Laplace method), then $\mathrm{I}\mathrm{n}\mathrm{d}(-{S_{x^{″}x}}^{″}(x^0))=0$.

• If $S(x)$ is purely imaginary for real x (i.e., $\mathrm{\Re }[S(x)]=0,\mathrm{\forall }x\in {\mathbb{ℝ}}^n$) and $x^0\in {\mathbb{ℝ}}^n$ (aka, the multidimensional stationary phase method), then $\mathrm{I}\mathrm{n}\mathrm{d}(-{S_{x^{″}x}}^{″}(x^0))=\frac{\pi }{4}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}{S_{x^{″}x}}^{″}(x_0),$ where $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}{S_{x^{″}x}}^{″}(x_0)$ denotes the signature of matrix ${S_{x^{″}x}}^{″}(x_0)$, which equals to the number of negative eigenvalues minus the number of positive ones. It is noteworthy that in applications of the stationary phase method to the multidimensional WKB approximation in quantum mechanics (as well as in optics), $\mathrm{I}\mathrm{n}\mathrm{d}$ is related to the Maslov index see, e.g., Template:Harvtxt and Template:Harvtxt.

The case of multiple non-degenerate saddle points

If the function $S(x)$ has multiple isolated non-degenerate saddle points (i.e., $\mathrm{\nabla }S(x^{(k)})=0,$ $\det S^{\prime }{\text{'}}_{xx}(x^{(k)})\ne 0,$ $x^{(k)}\in {\mathrm{\Omega }}_x^{(k)},$ where $\{{\mathrm{\Omega }}_x^{(k)}{\}}_{k=1}^K$ is an open cover of ${\mathrm{\Omega }}_x$), calculation of the integral asymptotic is reduced to the case of a singe saddle point by employing the partition of unity.

The partition of unity allows us to construct a set of continuous functions ${\{{\rho }_k(x):{\mathrm{\Omega }}_x\to [0,1]\}}_{k=1}^K$ such that $\sum _{k=1}^K{\rho }_k(x)=1,\mathrm{\forall }x\in {\mathrm{\Omega }}_x,$ and each function ${\rho }_k(x)$ vanishes outside ${\mathrm{\Omega }}_x^{(k)}$. Whence,

$\begin{array}{rl}& \underset{I_x\subset {\mathrm{\Omega }}_x}{\int }f(x)e^{\lambda S(x)}dx\equiv \sum _{k=1}^K\underset{I_x\subset {\mathrm{\Omega }}_x}{\int }{\rho }_k(x)f(x)e^{\lambda S(x)}dx\\ & \stackrel{\lambda \to +\mathrm{\infty }}{\to }\sum _{k=1}^K\underset{\text{a neighborhood of }{x}^{(k)}}{\int }f(x)e^{\lambda S(x)}dx={\left(\frac{2\pi }{\lambda }\right)}^{n/2}\sum _{k=1}^K{e}^{\lambda S(x^{(k)})}{[\det (-{S_{x^{″}x}}^{″}(x^{(k)}))]}^{-1/2}f(x^{(k)}),\end{array}$

where equation (13) was utilized at the last stage, and the pre-exponential function $f(x)$ at least must be continuous.

The other cases

When $\mathrm{\nabla }S(z^0)=0$ and $\det S^{\prime }{\text{'}}_{zz}(z^0)=0$, the point $z^0\in {\mathbb{ℂ}}^n$ is called a degenerate saddle point of a function $S(z)$.

Calculating the asymptotic of $\int f(x)e^{\lambda S(x)}dx$, when $\lambda \to +\mathrm{\infty }$, $f(x)$ is continuous, and $S(z)$ has a degenerate saddle point, is a very rich problem, whose solution heavily relies on the catastrophe theory. Here, the catastrophe theory replaces the Morse lemma, valid only in the non-degenerate case, to transform the function $S(z)$ into one of the multitude of canonical representations. For further details see, e.g., Template:Harvtxt and Template:Harvtxt.

Integrals with degenerate saddle points naturally appear in many applications including optical caustics and the multidimensional WKB approximation in quantum mechanics.

The other cases such as, e.g., $f(x)$ and/or $S(x)$ are discontinuous or when an extremum of $S(x)$ lies at the integration region's boundary, require special care (see, e.g., Template:Harvtxt and Template:Harvtxt).

Extensions and generalizations

An extension of the steepest descent method is the so-called nonlinear stationary phase/steepest descent method. Here, instead of integrals, one needs to evaluate asymptotically solutions of Riemann–Hilbert factorization problems.

Given a contour C in the complex sphere, a function ƒ defined on that contour and a special point, say infinity, one seeks a function M holomorphic away from the contour C, with prescribed jump across C, and with a given normalization at infinity. If ƒ and hence M are matrices rather than scalars this is a problem that in general does not admit an explicit solution.

An asymptotic evaluation is then possible along the lines of the linear stationary phase/steepest descent method. The idea is to reduce asymptotically the solution of the given Riemann–Hilbert problem to that of a simpler, explicitly solvable, Riemann–Hilbert problem. Cauchy's theorem is used to justify deformations of the jump contour.

The nonlinear stationary phase was introduced by Deift and Zhou in 1993, based on earlier work of the Russian mathematician Alexander Its. A (properly speaking) nonlinear steepest descent method was introduced by Kamvissis, K. McLaughlin and P. Miller in 2003, based on previous work of Lax, Levermore, Deift, Venakides and Zhou. As in the linear case, steepest descent contours solve a min-max problem.

The nonlinear stationary phase/steepest descent method has applications to the theory of soliton equations and integrable models, random matrices and combinatorics.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 English translation in Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Template:Eom.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 [in Russian].
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Unpublished note, reproduced in Riemann's collected papers.)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.