查看“︁雅可比-安格展开式”︁的源代码

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在[[数学]]中，'''雅可比-安格展开式'''（{{lang-en|Jacobi-Anger Expansion}}），或称'''雅可比-安格恒等式'''（{{lang-en|Jacobi-Anger Identity}}），是一种将特定形式的复指数函数展开成无穷个谐波分量之和的方法，在物理学（例如在[[平面波]]和[[柱面波]]之间转换）等领域中有所应用。此展开式以19世纪数学家[[卡爾·雅可比|卡尔·雅可比]]和{{Tsl|en|Carl Theodor Anger}}的名字命名。

从信号学的角度看，雅可比-安格展开式能够将具有特定[[频率调制]]形式的复指数信号展开成无穷个频率为整数倍基频的谐波分量之和，因而可用于对频率调制进行谱线分析。{{FACT}}

==定义形式==

雅可比-安格展开式的定义形式如下：<ref name=Colton_Kress_32/><ref name=Cuyt_et_al_344/>

: <math>\mathrm{e}^{i z \cos \theta} \equiv \sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, \mathrm{e}^{i n \theta}</math>

其中<math>i</math>是[[虛數單位|虚数单位]]，有<math display="inline">i^2=-1</math>；<math>z \in \C, \, \theta \in \R</math>；<math>J_n(z)</math>是<math>n</math>阶[[贝塞尔函数|第一类贝塞尔函数]]，比如下面是其[[麦克劳林级数]]展开形式：{{FACT}}

: <math>J_n(z) = \sum_{k=0}^{+\infin} \left( \frac {(-1)^k}{k!(n+k)!} \left( \frac z 2 \right)^{n+2k} \right)</math>

上述形式的贝塞尔函数要求<math>n</math>为非负整数，这里通过加入负整数扩充定义<math>J_{-n}(z) = (-1)^n\, J_{n}(z)</math>，让<math>J_n(z)</math>对所有整数阶次<math>n</math>有效。

==其他形式==

===正弦形式===

通过将定义形式中的<math>\theta</math>用<math>\theta-\frac{\pi}{2}</math>替代，可得到雅可比-安格展开式的正弦形式：<ref name=Colton_Kress_32/><ref name=Cuyt_et_al_344/>

: <math>\mathrm{e}^{i z \sin \theta} \equiv \sum_{n=-\infty}^{\infty} J_n(z)\, \mathrm{e}^{i n \theta}</math>

===实值形式===
有时也会使用下面这些实值形式，它们是由原始公式和正弦形式经过[[欧拉公式]]变形得到的：<ref name="Abramowitz_Stegun_361"/>

:<math>
\begin{align}
  \cos(z \cos \theta) &\equiv J_0(z)+2 \sum_{n=1}^{\infty}(-1)^n J_{2n}(z) \cos(2n \theta),
  \\
  \sin(z \cos \theta) &\equiv -2 \sum_{n=1}^{\infty}(-1)^n J_{2n-1}(z) \cos\left[\left(2n-1\right) \theta\right],
  \\
  \cos(z \sin \theta) &\equiv J_0(z)+2 \sum_{n=1}^{\infty} J_{2n}(z) \cos(2n \theta),
  \\
  \sin(z \sin \theta) &\equiv 2 \sum_{ n=1 }^{\infty} J_{2n-1}(z) \sin\left[\left(2n-1\right) \theta\right].
\end{align}
</math>

==参考文献==

{{reflist|refs=
<ref name="Colton_Kress_32">{{ citation
 | title=Inverse acoustic and electromagnetic scattering theory
 | series=Applied Mathematical Sciences
 | volume=93
 | last1=Colton
 | first1=David
 | last2=Kress
 | first2=Rainer
 | edition=2nd
 | year=1998
 | isbn=978-3-540-62838-5
 | page=32
}}</ref>

<ref name="Cuyt_et_al_344">{{ citation
 | title=Handbook of continued fractions for special functions
 | first1=Annie
 | last1=Cuyt
 | first2=Vigdis
 | last2=Petersen
 | first3=Brigitte
 | last3=Verdonk
 | first4=Haakon
 | last4=Waadeland
 | first5=William B.
 | last5=Jones
 | publisher=Springer
 | year=2008
 | isbn=978-1-4020-6948-2
 | page=344
}}</ref>

<ref name="Abramowitz_Stegun_361">{{cite book
|editor-first1=Milton
|editor-last1=Abramowitz
|editor-first2=Irene Ann
|editor-last2=Stegun
|author1={{{author|{{{author1|}}}}}}
|author2={{{author2|}}}
|author-first1={{{author-first|{{{author-first1|{{{author1-first|{{{first|{{{first1|}}}}}}}}}}}}}}}
|author-last1={{{author-last|{{{author-last1|{{{author1-last|{{{last|{{{last1|}}}}}}}}}}}}}}}
|author-first2={{{author-first2|{{{author2-first|{{{first2|}}}}}}}}}
|author-last2={{{author-last2|{{{author2-last|{{{last2|}}}}}}}}}
|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
|location=Washington D.C.<!-- NBS edition -->; New York<!-- Dover edition -->
|publisher=United States Department of Commerce, National Bureau of Standards; Dover Publications
|series=Applied Mathematics Series
|volume=55
|page=361
|date=1983<!-- Dover 9th reprint edition carries no date indication, but samples from around 1983 for USD17.95 as well as copies with ISBN-13, introduced after 2005, are known to exist and have the same set of additional errata. Hence took the oldest known date. -->
|edition=Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first<!-- edition -->
|isbn=978-0-486-61272-0<!-- of 9th Dover reprint -->
|lccn=64-60036<!-- original Government edition 1964, 1st printing -->
|id={{LCCN|6512253}}<!-- Dover reprint edition 1965, 1st printing -->
|mr=0167642
|contribution={{#if:{{{1|}}}|Chapter {{{1|}}}}}
|pages={{{2|}}}{{#if:{{{4|}}}|, {{{4|}}}}}
|contribution-url={{#if:{{{1|}}}|{{#if:{{{2|}}}|http://www.math.ubc.ca/~cbm/aands/page_{{{2|}}}.htm}}}}}}{{#if:{{{3|}}}|{{#if:{{{4|}}}| &ensp;See also [http://www.math.ubc.ca/~cbm/aands/page_{{{4}}}.htm chapter {{{3|}}}]|}}.}}</ref>}}

[[Category:数学恒等式]]