Partial fractions in complex analysis: Difference between revisions

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{{Short description|Way of writing a meromorphic function}}
In [[complex analysis]], a '''partial fraction expansion''' is a way of writing a [[meromorphic function]] ''<math>f(z)''</math> as an infinite sum of [[rational functions]] and [[polynomials]]. When ''<math>f(z)''</math> is a rational function, this reduces to the usual [[partial fractions|method of partial fractions]].

==Motivation==

By using [[polynomial long division]] and the partial fraction technique from algebra, any rational function can be written as a sum of terms of the form ''1 /<math display=inline>\frac{1}{(az + b)^{^k}''} + ''p(z)''</math>, where ''<math>a''</math> and ''<math>b''</math> are complex, ''<math>k''</math> is an integer, and ''<math>p(z)''</math> is a polynomial . Just as [[polynomial factorization]] can be generalized to the [[Weierstrass factorization theorem]], there is an analogy to partial fraction expansions for certain meromorphic functions.

A proper rational function, i.e. (one for which the [[degree of a polynomial|degree]] of the denominator is greater than the degree of the numerator,) has a partial fraction expansion with no polynomial terms. Similarly, a meromorphic function ''<math>f(z)''</math> for which <math>|''f(z)''|</math> goes to 0 as ''<math>z''</math> goes to infinity at least as quickly as <math display=inline>|''\frac{1/}{z''}|,</math> has an expansion with no polynomial terms.

==Calculation==

Let ''<math>f(z)''</math> be a function meromorphic in the finite complex plane with [[pole (complex analysis)|poles]] at ''&lambda;<submath>1</sub>''\lambda_1, ''&lambda;₂''\lambda_2, ...,</math> and let (''&Gamma;<submath>1</sub>''(\Gamma_1, ''&Gamma;₂''\Gamma_2, ...)</math> be a sequence of simple closed curves such that:

* The origin lies inside each curve ''&Gamma;<submath>k\Gamma_k</submath>''

* No curve passes through a pole of ''<math>f''</math>

* ''&Gamma;<submath>k\Gamma_k</submath>'' lies inside ''&Gamma;<submath>\Gamma_{k+1}</submath>'' for all ''<math>k''</math>

* <math>\lim_{k\rightarrow \infty} d(\Gamma_k) = \infty</math>, where ''<math>d(&Gamma;<sub>k\Gamma_k)</submath>)'' gives the distance from the curve to the origin

* one more condition of compatibility with the poles <math>\lambda_k</math>, described at the end of this section

Suppose also that there exists an integer ''<math>p''</math> such that

:<math>\lim_{k\rightarrow \infty} \oint_{\Gamma_k} \left|\frac{f(z)}{z^{p+1}}\right| |dz| < \infty</math>

Suppose also that there exists an integer ''p'' such that

Writing <math>\operatorname{PP}(''f(z)''; ''z = &lambda;<sub>k\lambda_k)</submath>'') for the [[principal part]] of the [[Laurent expansion]] of ''<math>f''</math> about the point ''&lambda;<submath>k\lambda_k</submath>'', we have

:<math>\lim_{k\rightarrow \infty} \oint_{\Gamma_k} \left|\frac{f(z)}{z^{p+1}}\right| dz < \infty</math>

Writing PP(''f(z)''; ''z = &lambda;_k'') for the [[principal part]] of the [[Laurent expansion]] of ''f'' about the point ''&lambda;_k'', we have

:<math>f(z) = \sum_{k=0}^{\infty} \operatorname{PP}(f(z); z = \lambda_k),</math>

if ''<math>p = -1'',</math>. andIf if ''<math>p > -1''</math>, then

:<math>f(z) = \sum_{k=0}^{\infty} (\operatorname{PP}(f(z); z = \lambda_k) + c_{0,k} + c_{1,k}z + \cdots + c_{p,k}z^p),</math>

where the coefficients ''c<submath>c_{j,k}</submath>'' are given by

:<math>c_{j,k} = \operatorname{Res}_{z=\lambda_k} \frac{f(z)}{z^{j+1}}</math>

&lambda;<submath>0\lambda_0</submath> should be set to 0, because even if ''<math>f(z)''</math> itself does not have a pole at 0, the residues[[residue (complex analysis)|residue]]s of ''<math display=inline>\frac{f(z)/}{z<sup>^{j+1}}</supmath>'' at ''<math>z'' = 0</math> must still be included in the sum.

Note that in the case of &lambda;<sub>0</submath>\lambda_0 = 0</math>, we can use the Laurent expansion of ''<math>f(z)''</math> about the origin to get

:<math>f(z) = \frac{a_{-m}}{z^m} + \frac{a_{-m+1}}{z^{m-1}} + \cdots + a_0 + a_1 z + \cdots</math>

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:<math>\sum_{j=0}^p c_{j,k}z^j = a_0 + a_1 z + \cdots + a_p z^p</math>

so that the polynomial terms contributed are exactly the [[regular part]] of the Laurent series up to ''z<supmath>z^p</supmath>''.

For the other poles ''&lambda;<submath>k\lambda_k</submath>'' where ''<math>k'' &\ge; 1</math>, ''1/z<supmath display=inline>\frac{1}{z^{j+1}}</supmath>'' can be pulled out of the [[Residue_residue (complex_analysiscomplex analysis)|residue]] calculations:

:<math>c_{j,k} = \frac{1}{\lambda_k^{j+1}} \operatorname{Res}_{z=\lambda_k} f(z)</math>

:<math>\sum_{j=0}^p c_{j,k}z^j = [\operatorname{Res}_{z=\lambda_k} f(z)] \sum_{j=0}^p \frac{1}{\lambda_k^{j+1}} z^j</math>

* To avoid issues with convergence, the poles should be ordered so that if &lambda;<submath>k\lambda_k</submath> is inside &Gamma;<submath>n\Gamma_n</submath>, then &lambda;<submath>j\lambda_j</submath> is also inside &Gamma;<submath>n\Gamma_n</submath> for all ''<math>j'' < ''k''</math>.

To avoid issues with convergence, the poles should be ordered so that if &lambda;_k is inside &Gamma;_n, then &lambda;_j is also inside &Gamma;_n for all ''j'' < ''k''.

==Example==

The simplest examples of meromorphic functions with an infinite number of poles are the non-[[entire function|entire]] trigonometric functions,. soAs takean the functionexample, <math>\tan(''z''). tan(''z'')</math> is meromorphic with poles at ''<math display=inline>(n + \frac{1/}{2})&\pi;''</math>, ''<math>n'' = 0, &plusmn;\pm 1, &plusmn;\pm 2, ...</math> The contours ''&Gamma;<submath>k\Gamma_k</submath>'' will be squares with vertices at ''&plusmn;&<math>\pm \pi; k &plusmn;\pm &\pi;ki'' k i</math> traversed counterclockwise, ''<math>k'' > 1</math>, which are easily seen to satisfy the necessary conditions.

[[Image:ContourSquares.PNG|right]]

The simplest examples of meromorphic functions with an infinite number of poles are the non-entire trigonometric functions, so take the function tan(''z''). tan(''z'') is meromorphic with poles at ''(n + 1/2)&pi;'', ''n'' = 0, &plusmn;1, &plusmn;2, ... The contours ''&Gamma;_k'' will be squares with vertices at ''&plusmn;&pi;k &plusmn; &pi;ki'' traversed counterclockwise, ''k'' > 1, which are easily seen to satisfy the necessary conditions.

On the horizontal sides of ''&Gamma;<submath>k\Gamma_k</submath>'',

:<math>z = t \pm \pi k i,\ \ t \in [-\pi k, \pi k],</math>

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:<math>|\tan(z)|^2 = \frac{\sin^2(t)\cosh^2(\pi k) + \cos^2(t)\sinh^2(\pi k)}{\cos^2(t)\cosh^2(\pi k) + \sin^2(t)\sinh^2(\pi k)}</math>

<math>\sinh(''x'') < \cosh(''x'')</math> for all real ''<math>x''</math>, which yields

:<math>|\tan(z)|^2 < \frac{\cosh^2(\pi k)(\sin^2(t) + \cos^2(t))}{\sinh^2(\pi k)(\cos^2(t) + \sin^2(t))} = \coth^2(\pi k)</math>

For ''<math>x'' > 0</math>, <math>\coth(''x'')</math> is continuous, decreasing, and bounded below by 1, so it follows that on the horizontal sides of ''&Gamma;<submath>k\Gamma_k</submath>'', <math>|\tan(''z'')| < \coth(''&\pi;'')</math>. Similarly, it can be shown that <math>|\tan(''z'')| < 1</math> on the vertical sides of ''&Gamma;<submath>k\Gamma_k</submath>''.

With this bound on <math>|\tan(''z'')|</math> we can see that

:<math>\oint_{\Gamma_k} \left|\frac{\tan(z)}{z}\right| dz \le \operatorname{length}(\Gamma_k) \max_{z\in \Gamma_k} \left|\frac{\tan(z)}{z}\right| < 8k \pi \frac{\coth(\pi)}{k\pi} = 8\coth(\pi) < \infty.</math>

(TheThat is, the maximum of <math display=inline>|\frac{1/''}{z''}|</math> on ''&Gamma;<submath>k\Gamma_k</submath>'' occurs at the minimum of <math>|''z''|</math>, which is ''<math>k&\pi;'')</math>.

Therefore ''<math>p'' = 0</math>, and the partial fraction expansion of <math>\tan(''z'')</math> looks like

:<math>\tan(z) = \sum_{k=0}^{\infty} (\operatorname{PP}(\tan(z); z = \lambda_k) + \operatorname{Res}_{z=\lambda_k} \frac{\tan(z)}{z}).</math>

The principal parts and residues[[residue (complex analysis)|residue]]s are easy enough to calculate, as all the poles of <math>\tan(''z'')</math> are simple and have residue -1:

:<math>\operatorname{PP}(\tan(z); z = (n + \frac{1}{2})\pi) = \frac{-1}{z - (n + \frac{1}{2})\pi}</math>

:<math>\operatorname{Res}_{z=(n + \frac{1}{2})\pi} \frac{\tan(z)}{z} = \frac{-1}{(n + \frac{1}{2})\pi}</math>

We can ignore ''&lambda;<submath>0</sub>''\lambda_0 = 0</math>, since both <math>\tan(''z'')</math> and <math display=inline>\frac{\tan(''z'')/''}{z''}</math> are analytic at 0, so there is no contribution to the sum, and ordering the poles ''&lambda;<submath>k\lambda_k</submath>'' so that ''&lambda;<submath display=inline>1</sub>''\lambda_1 = ''&\frac{\pi;''/}{2}, ''&lambda;₂''\lambda_2 = \frac{-''&\pi;''/}{2}, ''&lambda;₃''\lambda_3 = \frac{3''&\pi;''}{2}</2math>, etc., gives

:<math>\tan(z) = \sum_{k=0}^{\infty} \left[\left(\frac{-1}{z - (k + \frac{1}{2})\pi} - \frac{1}{(k + \frac{1}{2})\pi}\right) + \left(\frac{-1}{z + (k + \frac{1}{2})\pi} + \frac{1}{(k + \frac{1}{2})\pi}\right)\right]</math>

:<math>\tan(z) = \sum_{k=0}^{\infty} \frac{-2z}{z^2 - (k + \frac{1}{2})^2\pi^2}</math>

==Applications==

==Application to infinite products==

===Infinite products===

Because the partial fraction expansion often yields sums of ''1/(a+bz)'', it can be useful in finding a way to write a function as an [[infinite product]]; integrating both sides gives a sum of logarithms, and exponentiating gives the desired product:

Because the partial fraction expansion often yields sums of ''<math display=inline>\frac{1/(}{a+bz)''}</math>, it can be useful in finding a way to write a function as an [[infinite product]]; integrating both sides gives a sum of logarithms, and exponentiating gives the desired product:

:<math>\tan(z) = -\sum_{k=0}^{\infty} \left(\frac{1}{z - (k + \frac{1}{2})\pi} + \frac{1}{z + (k + \frac{1}{2})\pi}\right)</math>

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:<math>\cos z = \prod_{k=0}^{\infty} \left(1 - \frac{z^2}{(k + \frac{1}{2})^2\pi^2}\right).</math>

===Laurent series===

The partial fraction expansion for a function can also be used to find a Laurent series for it by simply replacing the rational functions in the sum with their Laurent series, which are often not difficult to write in closed form. This can also lead to interesting identities if a Laurent series is already known.

Recall that

:<math>\tan(z) = \sum_{k=0}^{\infty} \frac{-2z}{z^2 - (k + \frac{1}{2})^2\pi^2} = \sum_{k=0}^{\infty} \frac{-8z}{4z^2 - (2k + 1)^2\pi^2}.</math>

We can expand the summand using a geometric series:

:<math>\frac{-8z}{4z^2 - (2k + 1)^2\pi^2} = \frac{8z}{(2k + 1)^2\pi^2} \frac{1}{1 - (\frac{2z}{(2k + 1)\pi})^2} = \frac{8}{(2k + 1)^2\pi^2}\sum_{n=0}^{\infty} \frac{2^{2n}}{(2k + 1)^{2n}\pi^{2n}} z^{2n + 1}.</math>

Substituting back,

:<math>\tan(z) = 2\sum_{k=0}^{\infty} \sum_{n=0}^{\infty} \frac{2^{2n+2}}{(2k + 1)^{2n+2}\pi^{2n+2}} z^{2n + 1},</math>

which shows that the coefficients <math>a_n</math> in the Laurent (Taylor) series of <math>\tan(z)</math> about <math>z = 0</math> are

:<math>a_{2n+1} = \frac{T_{2n+1}}{(2n+1)!} = \frac{2^{2n+3}}{\pi^{2n+2}} \sum_{k=0}^{\infty} \frac{1}{(2k + 1)^{2n+2}}</math>

:<math>a_{2n} = \frac{T_{2n}}{(2n)!} = 0,</math>

where <math>T_n</math> are the [[tangent numbers]].

Conversely, we can compare this formula to the Taylor expansion for <math>\tan(z)</math> about <math>z = 0</math> to calculate the infinite sums:

:<math>\tan(z) = z + \frac{1}{3}z^3 + \frac{2}{15}z^5 + \cdots</math>

:<math>\sum_{k=0}^{\infty} \frac{1}{(2k + 1)^2} = \frac{\pi^2}{2^3} = \frac{\pi^2}{8}</math>

:<math>\sum_{k=0}^{\infty} \frac{1}{(2k + 1)^4} = \frac{1}{3} \frac{\pi^4}{2^5} = \frac{\pi^4}{96}.</math>

==See also==

* [[Partial fraction]]s

* [[PathLine integral]]

* [[Residue (complex analysis)]]

* [[Residue theorem]]

==References==

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[[Category:Complex analysis]]

[[Category:Partial fractions]]