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| [[File:Parabolic cylindrical coordinates.png|thumb|right|350px|[[Coordinate system#Coordinate surface|Coordinate surfaces]] of parabolic cylindrical coordinates. Parabolic cylinder functions occur when [[separation of variables]] is used on [[Laplace equation|Laplace's equation]] in these coordinates]]
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| In [[mathematics]], the '''parabolic cylinder functions''' are [[special function]]s defined as solutions to the differential equation
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| :<math>\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0.</math>
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| This equation is found when the technique of [[separation of variables]] is used on [[Laplace equation|Laplace's equation]] when expressed in [[parabolic cylindrical coordinates]].
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| The above equation may be brought into two distinct forms (A) and (B) by [[completing the square]] and rescaling ''z'', called [[H. F. Weber]]'s equations {{harv|Weber|1869}}:
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| :<math>\frac{d^2f}{dz^2} - \left(\tfrac14z^2+a\right)f=0</math> (A)
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| and
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| :<math>\frac{d^2f}{dz^2} + \left(\tfrac14z^2-a\right)f=0.</math> (B)
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| If
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| :<math>f(a,z)\,</math>
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| is a solution, then so are
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| :<math>f(a,-z), f(-a,iz)\text{ and }f(-a,-iz).\,</math>
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| If
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| :<math>f(a,z)\,</math>
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| is a solution of equation (A), then
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| :<math>f(-ia,ze^{(1/4)\pi i})\,</math>
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| is a solution of (B), and, by symmetry,
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| :<math>f(-ia,-ze^{(1/4)\pi i}), f(ia,-ze^{-(1/4)\pi i})\text{ and }f(ia,ze^{-(1/4)\pi i})\,</math>
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| are also solutions of (B).
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| ==Solutions==
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| There are independent even and odd solutions of the form (A). These are given by (following the notation of [[Abramowitz and Stegun]] (1965)):
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| :<math>y_1(a;z) = \exp(-z^2/4) \;_1F_1
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| \left(\tfrac12a+\tfrac14; \;
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| \tfrac12\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{even})</math>
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| and
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| :<math>y_2(a;z) = z\exp(-z^2/4) \;_1F_1
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| \left(\tfrac12a+\tfrac34; \;
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| \tfrac32\; ; \; \frac{z^2}{2}\right)\,\,\,\,\,\, (\mathrm{odd})</math>
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| where <math>\;_1F_1 (a;b;z)=M(a;b;z)</math> is the [[confluent hypergeometric function]].
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| Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity:
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| :<math>
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| U(a,z)=\frac{1}{2^\xi\sqrt{\pi}}
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| \left[
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| \cos(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z)
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| -\sqrt{2}\sin(\xi\pi)\Gamma(1-\xi)\,y_2(a,z)
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| \right]
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| </math>
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| :<math>
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| V(a,z)=\frac{1}{2^\xi\sqrt{\pi}\Gamma[1/2-a]}
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| \left[
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| \sin(\xi\pi)\Gamma(1/2-\xi)\,y_1(a,z)
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| +\sqrt{2}\cos(\xi\pi)\Gamma(1-\xi)\,y_2(a,z)
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| \right]
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| </math>
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| where
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| :<math>
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| \xi=\frac{1}{2}a+\frac{1}{4} .
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| </math> | |
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| The function ''U''(''a'', ''z'') approaches zero for large values of |z| and |arg(''z'')| < π/2, while ''V''(''a'', ''z'') diverges for large values of positive real ''z'' .
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| :<math> | |
| \lim_{|z|\rightarrow\infty}U(a,z)/e^{-z^2/4}z^{-a-1/2}=1\,\,\,\,(\text{for}\,|\arg(z)|<\pi/2)
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| </math>
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| and
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| :<math> | |
| \lim_{|z|\rightarrow\infty}V(a,z)/\sqrt{\frac{2}{\pi}}e^{z^2/4}z^{a-1/2}=1\,\,\,\,(\text{for}\,\arg(z)=0) .
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| </math>
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| For [[half-integer]] values of ''a'', these (that is, ''U'' and ''V'') can be re-expressed in terms of [[Hermite polynomials]]; alternatively, they can also be expressed in terms of [[Bessel function]]s.
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| The functions ''U'' and ''V'' can also be related to the functions ''D<sub>p</sub>''(''x'') (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)):
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| :<math>U(a,x)=D_{-a-\tfrac12}(x),</math> | |
| :<math>V(a,x)=\frac{\Gamma(\tfrac12+a)}{\pi}[\sin( \pi a) D_{-a-\tfrac12}(x)+D_{-a-\tfrac12}(-x)] .</math>
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| {{no footnotes|date=December 2010}}
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| == References ==
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| * {{AS ref |19|686}}
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| *{{springer|id=W/w097310|title=Weber equation|first=N.Kh.|last= Rozov}}
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| *{{dlmf|id=12|first=N. M. |last=Temme}}
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| *Weber, H.F. (1869) "Ueber die Integration der partiellen Differentialgleichung <math>\partial^2u/\partial x^2+\partial^2u/\partial y^2+k^2u=0</math>". ''Math. Ann.'', 1, 1–36
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| *Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" ''Proc. London Math. Soc.''35, 417–427.
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| {{DEFAULTSORT:Parabolic Cylinder Function}}
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| [[Category:Special hypergeometric functions]]
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I'm Curt and I live in Genthin.
I'm interested in Comparative Politics, Creative writing and Vietnamese art. I like travelling and reading fantasy.
my website :: FIFA Coin Generator