Residue theorem

 ( Theorems In Complex Analysis ) 

In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.

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Statement of Cauchy's residue theorem The statement is as follows:

Residue theorem: Let $U$ be a simply connected open subset of the complex plane containing a finite list of points $a\_1,\; \backslash ldots,\; a\_n,$ $U\_0\; =\; U\; \backslash smallsetminus\; \backslash \{a\_1,\; \backslash ldots,\; a\_n\backslash \},$ and a function $f$ holomorphic on $U\_0.$ Letting $\backslash gamma$ be a closed rectifiable curve in $U\_0,$ and denoting the residue of $f$ at each point $a\_k$ by $\backslash operatorname\{Res\}(f,\; a\_k)$ and the winding number of $\backslash gamma$ around $a\_k$ by $\backslash operatorname\{I\}(\backslash gamma,\; a\_k),$ the line integral of $f$ around $\backslash gamma$ is equal to $2\backslash pi\; i$ times the sum of residues, each counted as many times as $\backslash gamma$ winds around the respective point:

$$\backslash oint\_\backslash gamma\; f(z)\backslash ,\; dz\; =\; 2\backslash pi\; i\; \backslash sum\_\{k=1\}^n\; \backslash operatorname\{I\}(\backslash gamma,\; a\_k)\; \backslash operatorname\{Res\}(f,\; a\_k).$$

If $\backslash gamma$ is a positively oriented Jordan curve, $\backslash operatorname\{I\}(\backslash gamma,\; a\_k)$ is $1$ if $a\_k$ is in the interior of $\backslash gamma$ and $0$ if not, therefore

$$\backslash oint\_\backslash gamma\; f(z)\backslash ,\; dz\; =\; 2\backslash pi\; i\; \backslash sum\; \backslash operatorname\{Res\}(f,\; a\_k)$$

with the sum over those $a\_k$ inside

The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve must first be reduced to a set of simple closed curves $\backslash \{\backslash gamma\_i\backslash \}$ whose total is equivalent to $\backslash gamma$ for integration purposes; this reduces the problem to finding the integral of $f\backslash ,\; dz$ along a Jordan curve $\backslash gamma\_i$ with interior $V.$ The requirement that $f$ be holomorphic on $U\_0\; =\; U\; \backslash smallsetminus\; \backslash \{a\_k\backslash \}$ is equivalent to the statement that the exterior derivative $d(f\backslash ,\; dz)\; =\; 0$ on $U\_0.$ Thus if two planar regions $V$ and $W$ of $U$ enclose the same subset $\backslash \{a\_j\backslash \}$ of $\backslash \{a\_k\backslash \},$ the regions $V\; \backslash smallsetminus\; W$ and $W\; \backslash smallsetminus\; V$ lie entirely in $U\_0,$ hence

$$\backslash int\_\{V\; \backslash smallsetminus\; W\}\; d(f\; \backslash ,\; dz)\; -\; \backslash int\_\{W\; \backslash smallsetminus\; V\}\; d(f\; \backslash ,\; dz)$$

is well-defined and equal to zero. Consequently, the contour integral of $f\backslash ,\; dz$ along $\backslash gamma\_j\; =\; \backslash partial\; V$ is equal to the sum of a set of integrals along paths $\backslash gamma\_j,$ each enclosing an arbitrarily small region around a single $a\_j$ — the residues of $f$ (up to the conventional factor $2\backslash pi\; i$ at $\backslash \{a\_j\backslash \}.$ Summing over $\backslash \{\backslash gamma\_j\backslash \},$ we recover the final expression of the contour integral in terms of the winding numbers $\backslash \{\backslash operatorname\{I\}(\backslash gamma,\; a\_k)\backslash \}.$

In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.

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Calculation of residues

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Examples

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An integral along the real axis The integral $$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{e^\{itx\}\}\{x^2+1\}\backslash ,dx$$

arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of .

Suppose and define the contour that goes along the real number line from to and then counterclockwise along a semicircle centered at 0 from to . Take to be greater than 1, so that the imaginary number unit is enclosed within the curve. Now consider the contour integral $$\backslash int\_C\; \{f(z)\}\backslash ,dz\; =\; \backslash int\_C\; \backslash frac\{e^\{itz\}\}\{z^2+1\}\backslash ,dz.$$

Since is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator is zero. Since , that happens only where or . Only one of those points is in the region bounded by this contour. Because is the residue of at is $$\backslash operatorname\{Res\}\_\{z=i\}f(z)=\backslash frac\{e^\{-t\}\}\{2i\}.$$

According to the residue theorem, then, we have $$\backslash int\_C\; f(z)\backslash ,dz=2\backslash pi\; i\backslash cdot\backslash operatorname\{Res\}\backslash limits\_\{z=i\}f(z)=2\backslash pi\; i\; \backslash frac\{e^\{-t\}\}\{2i\}\; =\; \backslash pi\; e^\{-t\}.$$

The contour may be split into a straight part and a curved arc, so that $$\backslash int\_\{\backslash mathrm\{straight\}\}\; f(z)\backslash ,dz+\backslash int\_\{\backslash mathrm\{arc\}\}\; f(z)\backslash ,dz=\backslash pi\; e^\{-t\}$$ and thus $$\backslash int\_\{-a\}^a\; f(z)\backslash ,dz\; =\backslash pi\; e^\{-t\}-\backslash int\_\{\backslash mathrm\{arc\}\}\; f(z)\backslash ,dz.$$

Using some Estimation lemma, we have $$\backslash left|\backslash int\_\{\backslash mathrm\{arc\}\}\backslash frac\{e^\{itz\}\}\{z^2+1\}\backslash ,dz\backslash right|\; \backslash leq\; \backslash pi\; a\; \backslash cdot\; \backslash sup\_\{\backslash text\{arc\}\}\; \backslash left|\; \backslash frac\{e^\{itz\}\}\{z^2+1\}\; \backslash right|\; \backslash leq\; \backslash pi\; a\; \backslash cdot\; \backslash sup\_\{\backslash text\{arc\}\}\; \backslash frac\{1\}$$

\leq \frac{\pi a}{a^2 - 1}, and $$\backslash lim\_\{a\; \backslash to\; \backslash infty\}\; \backslash frac\{\backslash pi\; a\}\{a^2-1\}\; =\; 0.$$

The estimate on the numerator follows since , and for along the arc (which lies in the upper half-plane), the argument of lies between 0 and . So, $$\backslash left|e^\{itz\}\backslash right|\; =\; \backslash left|e^\{it|z|(\backslash cos\backslash varphi\; +\; i\backslash sin\backslash varphi)\}\backslash right|=\backslash left|e^\{-t|z|\backslash sin\backslash varphi\; +\; it|z|\backslash cos\backslash varphi\}\backslash right|=e^\{-t|z|\; \backslash sin\backslash varphi\}\; \backslash le\; 1.$$

Therefore, $$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{e^\{itz\}\}\{z^2+1\}\backslash ,dz=\backslash pi\; e^\{-t\}.$$

If then a similar argument with an arc that winds around rather than shows that

$$\backslash int\_\{-\backslash infty\}^\backslash infty\backslash frac\{e^\{itz\}\}\{z^2+1\}\backslash ,dz=\backslash pi\; e^t,$$

and finally we have $$\backslash int\_\{-\backslash infty\}^\backslash infty\backslash frac\{e^\{itz\}\}\{z^2+1\}\backslash ,dz=\backslash pi\; e^\{-\backslash left|t\backslash right|\}.$$

(If then the integral yields immediately to elementary calculus methods and its value is .)

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Evaluating zeta functions The fact that has simple poles with residue 1 at each integer can be used to compute the sum $$\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; f(n).$$

Consider, for example, . Let be the rectangle that is the boundary of with positive orientation, with an integer . By the residue formula,

$$\backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_\{\backslash Gamma\_N\}\; f(z)\; \backslash pi\; \backslash cot(\backslash pi\; z)\; \backslash ,\; dz\; =\; \backslash operatorname\{Res\}\backslash limits\_\{z\; =\; 0\}\; +\; \backslash sum\_\{n\; =\; -N\; \backslash atop\; n\backslash ne\; 0\}^N\; n^\{-2\}.$$

The left-hand side goes to zero as since $|\backslash cot(\backslash pi\; z)|$ is uniformly bounded on the contour, thanks to using $x\; =\; \backslash pm\; \backslash left(\backslash frac\; 12\; +\; N\backslash right)$ on the left and right side of the contour, and so the integrand has order $O(N^\{-2\})$ over the entire contour. On the other hand,. Note that the Bernoulli number $B\_\{2n\}$ is denoted by $B\_\{n\}$ in Whittaker & Watson's book.

$$\backslash frac\{z\}\{2\}\; \backslash cot\backslash left(\backslash frac\{z\}\{2\}\backslash right)\; =\; 1\; -\; B\_2\; \backslash frac\{z^2\}\{2!\}\; +\; \backslash cdots$$ where the Bernoulli number $B\_2\; =\; \backslash frac\{1\}\{6\}.$

(In fact, .) Thus, the residue is . We conclude:

$$\backslash sum\_\{n\; =\; 1\}^\backslash infty\; \backslash frac\{1\}\{n^2\}\; =\; \backslash frac\{\backslash pi^2\}\{6\}$$ which is a proof of the Basel problem.

The same argument works for all $f(x)\; =\; x^\{-2n\}$ where $n$ is a positive integer, giving us$$\backslash zeta(2n)\; =\; \backslash frac\{(-1)^\{n+1\}B\_\{2n\}(2\backslash pi)^\{2n\}\}\{2(2n)!\}.$$The trick does not work when $f(x)\; =\; x^\{-2n-1\}$, since in this case, the residue at zero vanishes, and we obtain the useless identity $0\; +\; \backslash zeta(2n+1)\; -\; \backslash zeta(2n+1)\; =\; 0$.

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Evaluating Eisenstein series The same trick can be used to establish the sum of the Eisenstein series:$$\backslash pi\; \backslash cot(\backslash pi\; z)\; =\; \backslash lim\_\{N\; \backslash to\; \backslash infty\}\; \backslash sum\_\{n=-N\}^N\; (z\; -\; n)^\{-1\}.$$

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See also

• Residue (complex analysis)
• Cauchy's integral formula
• Glasser's master theorem
• Jordan's lemma
• Methods of contour integration
• Morera's theorem
• Nachbin's theorem
• Residue at infinity
• Logarithmic form

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Notes

• Lars Ahlfors (1979). 9780070850088, McGraw Hill. ISBN 9780070850088
• Ernst L. Lindelöf (2025). 9782876470606, Editions Jacques Gabay. ISBN 9782876470606
• (1984). 9789027716231, D. Reidel Publishing Company. ISBN 9789027716231

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External links

• Residue theorem in MathWorld

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