Jump to navigation Jump to search

General

Display information for equation id:math.218677.1 on revision:218677

* Page found: Optical aberration (eq math.218677.1)

(force rerendering)

Cannot find the equation data in the database. Fetching from revision text.

Occurrences on the following pages:

Hash: ba6275457a491e6b480434106e4cce9b

TeX (original user input):

Z^{-m}_n(\rho,\phi) = R^m_n(\rho)\,\sin(m\,\phi), \!

TeX (checked):

Z_{n}^{-m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\sin(m\,\phi ),\!

LaTeXML (experimental; uses MathML) rendering

MathML (5.769 KB / 1.028 KB) :

Z n - m ( ρ , ϕ ) = R n m ( ρ ) sin ( m ϕ ) , superscript subscript Z n m ρ ϕ superscript subscript R n m ρ m ϕ {\displaystyle Z_{n}^{{-m}}(\rho,\phi)=R_{n}^{m}(\rho)\,\sin(m\,\phi),\!}
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle Z_{n}^{{-m}}(\rho,\phi)=R_{n}^{m}(\rho)\,\sin(m\,\phi),\!}" display="inline">
  <semantics id="p1.1.m1.1a">
    <mrow id="p1.1.m1.1.25" xref="p1.1.m1.1.25.cmml">
      <mrow id="p1.1.m1.1.25a" xref="p1.1.m1.1.25.cmml">
        <mrow id="p1.1.m1.1.25.1" xref="p1.1.m1.1.25.1.cmml">
          <msubsup id="p1.1.m1.1.25.1.2" xref="p1.1.m1.1.25.1.2.cmml">
            <mi id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">Z</mi>
            <mi id="p1.1.m1.1.2.1" xref="p1.1.m1.1.2.1.cmml">n</mi>
            <mrow id="p1.1.m1.1.3.1" xref="p1.1.m1.1.3.1.cmml">
              <mo id="p1.1.m1.1.3.1.1" xref="p1.1.m1.1.3.1.1.cmml">-</mo>
              <mi id="p1.1.m1.1.3.1.2" xref="p1.1.m1.1.3.1.2.cmml">m</mi>
            </mrow>
          </msubsup>
          <mo id="p1.1.m1.1.25.1.1" xref="p1.1.m1.1.25.1.1.cmml"></mo>
          <mrow id="p1.1.m1.1.25.1.3" xref="p1.1.m1.1.25.1.3.1.cmml">
            <mo stretchy="false" id="p1.1.m1.1.4">(</mo>
            <mi id="p1.1.m1.1.5" xref="p1.1.m1.1.5.cmml">ρ</mi>
            <mo id="p1.1.m1.1.6">,</mo>
            <mi id="p1.1.m1.1.7" xref="p1.1.m1.1.7.cmml">ϕ</mi>
            <mo stretchy="false" id="p1.1.m1.1.8">)</mo>
          </mrow>
        </mrow>
        <mo id="p1.1.m1.1.9" xref="p1.1.m1.1.9.cmml">=</mo>
        <mrow id="p1.1.m1.1.25.2" xref="p1.1.m1.1.25.2.cmml">
          <msubsup id="p1.1.m1.1.25.2.2" xref="p1.1.m1.1.25.2.2.cmml">
            <mi id="p1.1.m1.1.10" xref="p1.1.m1.1.10.cmml">R</mi>
            <mi id="p1.1.m1.1.11.1" xref="p1.1.m1.1.11.1.cmml">n</mi>
            <mi id="p1.1.m1.1.12.1" xref="p1.1.m1.1.12.1.cmml">m</mi>
          </msubsup>
          <mo id="p1.1.m1.1.25.2.1" xref="p1.1.m1.1.25.2.1.cmml"></mo>
          <mrow id="p1.1.m1.1.25.2.3">
            <mo stretchy="false" id="p1.1.m1.1.13">(</mo>
            <mi id="p1.1.m1.1.14" xref="p1.1.m1.1.14.cmml">ρ</mi>
            <mo rspace="4.2pt" stretchy="false" id="p1.1.m1.1.15">)</mo>
          </mrow>
          <mo id="p1.1.m1.1.25.2.1a" xref="p1.1.m1.1.25.2.1.cmml"></mo>
          <mrow id="p1.1.m1.1.25.2.4.2" xref="p1.1.m1.1.25.2.4.1.cmml">
            <mi id="p1.1.m1.1.17" xref="p1.1.m1.1.17.cmml">sin</mi>
            <mo id="p1.1.m1.1.25.2.4.2a" xref="p1.1.m1.1.25.2.4.1.cmml"></mo>
            <mrow id="p1.1.m1.1.25.2.4" xref="p1.1.m1.1.25.2.4.1.cmml">
              <mo stretchy="false" id="p1.1.m1.1.18" xref="p1.1.m1.1.25.2.4.1.cmml">(</mo>
              <mrow id="p1.1.m1.1.25.2.4.2.1" xref="p1.1.m1.1.25.2.4.1.cmml">
                <mpadded width="+1.7pt" id="p1.1.m1.1.19" xref="p1.1.m1.1.25.2.4.1.cmml">
                  <mi id="p1.1.m1.1.19a" xref="p1.1.m1.1.19.cmml">m</mi>
                </mpadded>
                <mo id="p1.1.m1.1.25.2.4.2.1.1" xref="p1.1.m1.1.25.2.4.2.1.1.cmml"></mo>
                <mi id="p1.1.m1.1.21" xref="p1.1.m1.1.21.cmml">ϕ</mi>
              </mrow>
              <mo stretchy="false" id="p1.1.m1.1.22" xref="p1.1.m1.1.25.2.4.1.cmml">)</mo>
            </mrow>
          </mrow>
        </mrow>
      </mrow>
      <mo id="p1.1.m1.1.25b" xref="p1.1.m1.1.25.cmml">,</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b">
      <apply id="p1.1.m1.1.25.cmml" xref="p1.1.m1.1.25">
        <eq id="p1.1.m1.1.9.cmml" xref="p1.1.m1.1.9"/>
        <apply id="p1.1.m1.1.25.1.cmml" xref="p1.1.m1.1.25.1">
          <times id="p1.1.m1.1.25.1.1.cmml" xref="p1.1.m1.1.25.1.1"/>
          <apply id="p1.1.m1.1.25.1.2.cmml" xref="p1.1.m1.1.25.1.2">
            <csymbol cd="ambiguous" id="p1.1.m1.1.25.1.2.1.cmml">superscript</csymbol>
            <apply id="p1.1.m1.1.25.1.2.2.cmml">
              <csymbol cd="ambiguous" id="p1.1.m1.1.25.1.2.2.1.cmml">subscript</csymbol>
              <ci id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">Z</ci>
              <ci id="p1.1.m1.1.2.1.cmml" xref="p1.1.m1.1.2.1">n</ci>
            </apply>
            <apply id="p1.1.m1.1.3.1.cmml" xref="p1.1.m1.1.3.1">
              <minus id="p1.1.m1.1.3.1.1.cmml" xref="p1.1.m1.1.3.1.1"/>
              <ci id="p1.1.m1.1.3.1.2.cmml" xref="p1.1.m1.1.3.1.2">m</ci>
            </apply>
          </apply>
          <interval closure="open" id="p1.1.m1.1.25.1.3.1.cmml" xref="p1.1.m1.1.25.1.3">
            <ci id="p1.1.m1.1.5.cmml" xref="p1.1.m1.1.5">ρ</ci>
            <ci id="p1.1.m1.1.7.cmml" xref="p1.1.m1.1.7">ϕ</ci>
          </interval>
        </apply>
        <apply id="p1.1.m1.1.25.2.cmml" xref="p1.1.m1.1.25.2">
          <times id="p1.1.m1.1.25.2.1.cmml" xref="p1.1.m1.1.25.2.1"/>
          <apply id="p1.1.m1.1.25.2.2.cmml" xref="p1.1.m1.1.25.2.2">
            <csymbol cd="ambiguous" id="p1.1.m1.1.25.2.2.1.cmml">superscript</csymbol>
            <apply id="p1.1.m1.1.25.2.2.2.cmml">
              <csymbol cd="ambiguous" id="p1.1.m1.1.25.2.2.2.1.cmml">subscript</csymbol>
              <ci id="p1.1.m1.1.10.cmml" xref="p1.1.m1.1.10">R</ci>
              <ci id="p1.1.m1.1.11.1.cmml" xref="p1.1.m1.1.11.1">n</ci>
            </apply>
            <ci id="p1.1.m1.1.12.1.cmml" xref="p1.1.m1.1.12.1">m</ci>
          </apply>
          <ci id="p1.1.m1.1.14.cmml" xref="p1.1.m1.1.14">ρ</ci>
          <apply id="p1.1.m1.1.25.2.4.1.cmml" xref="p1.1.m1.1.25.2.4.2">
            <sin id="p1.1.m1.1.17.cmml" xref="p1.1.m1.1.17"/>
            <apply id="p1.1.m1.1.25.2.4.2.1.cmml" xref="p1.1.m1.1.25.2.4.2">
              <times id="p1.1.m1.1.25.2.4.2.1.1.cmml" xref="p1.1.m1.1.25.2.4.2.1.1"/>
              <ci id="p1.1.m1.1.19.cmml" xref="p1.1.m1.1.19a">m</ci>
              <ci id="p1.1.m1.1.21.cmml" xref="p1.1.m1.1.21">ϕ</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle Z_{n}^{{-m}}(\rho,\phi)=R_{n}^{m}(\rho)\,\sin(m\,\phi),\!}</annotation>
  </semantics>
</math>

SVG image empty. Force Re-Rendering

SVG (11.455 KB / 4.055 KB) :

upper Z Subscript n Superscript negative m Baseline times left-parenthesis rho comma phi right-parenthesis equals upper R Subscript n Superscript m Baseline times left-parenthesis rho right-parenthesis times sine left-parenthesis m times phi right-parenthesis comma

MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools) rendering

MathML (0 B / 8 B) :

SVG image empty. Force Re-Rendering

SVG (0 B / 8 B) :


PNG (0 B / 8 B) :


Translations to Computer Algebra Systems

Translation to Maple

In Maple: (Z[n])^(- m)*(rho , phi)= (R[n])^(m)*(rho)* sin(m*phi),

Information about the conversion process:

\sin: Sine; Example: \sin@@{z}

Will be translated to: sin($0)

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E1

Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin


\phi: Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.

But this system don't know how to translate it as a constant. It was translated as a general letter.



Translation to Mathematica

In Mathematica: (Subscript[Z, n])^(- m)*(\[Rho], \[Phi])= (Subscript[R, n])^(m)*(\[Rho])* Sin[m*\[Phi]],

Information about the conversion process:

\sin: Sine; Example: \sin@@{z}

Will be translated to: Sin[$0]

Relevant links to definitions:

DLMF: http://dlmf.nist.gov/4.14#E1

Mathematica: https://reference.wolfram.com/language/ref/Sin.html


\phi: Could be the golden ratio == golden mean == golden section == extreme and mean ratio == medial section == divine proportion == divine section == golden proportion == golden cut == golden number.

But this system don't know how to translate it as a constant. It was translated as a general letter.



Similar pages

Calculated based on the variables occurring on the entire Optical aberration page

Identifiers

MathML observations

0results

0results

no statistics present please run the maintenance script ExtractFeatures.php

0 results

0 results