RESIDUE THEOREM

The 'residue theorem' in complex analysis is a powerful tool to evaluate line integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
The statement is as follows. Suppose ''U'' is a simply connected open subset of the complex plane 'C', ''a''₁,...,''a''_''n'' are finitely many points of ''U'' and ''f'' is a function which is defined and holomorphic on ''U'' {''a''₁,...,''a''_''n''}. If γ is a rectifiable curve in ''U'' which doesn't meet any of the points ''a''_''k'' and whose start point equals its endpoint, then
:$oint\_gamma\; f(z),\; dz\; =$
2pi i sum_{k=1}^n operatorname{I}(gamma, a_k)
operatorname{Res}( f, a_k ).
If γ is a Jordan curve, I(γ, ''a''_''k'') = 1 and so
:$oint\_gamma\; f(z),\; dz\; =$
2pi i sum_{k=1}^n operatorname{Res}( f, a_k ).
Here, Res(''f'', ''a''_''k'') denotes the residue of ''f'' at ''a''_''k'', and I(γ, ''a''_''k'') is the winding number of the curve γ about the point ''a''_''k''. This winding number is an integer which intuitively measures how often the curve γ winds around the point ''a''_''k''; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around ''a''_''k'' and 0 if γ doesn't move around ''a''_''k'' at all.
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. 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 if it is large enough, leaving only the real-axis part of the integral, the one we were originally interested in.

Contents

Example

See also

References

External links

Example

The integral
:$int\_\{-infty\}^infty\; \{e^\{itx\}\; over\; x^2+1\},dx$

(which arises in probability theory as (a scalar multiple
of) the characteristic function of the Cauchy distribution)
resists the techniques of elementary calculus. We will
evaluate it by expressing it as a limit of contour integrals
along the contour ''C'' that goes along the real
line from −''a'' to ''a'' and then counterclockwise along
a semicircle centered at 0 from ''a'' to −''a''. Take
''a'' to be greater than 1, so that the imaginary
unit ''i'' is enclosed within the curve. The contour integral is
:$int\_C\; \{e^\{itz\}\; over\; z^2+1\},dz.$
Since ''e''^''itz'' is an entire function
(having no singularities
at any point in the complex plane), this function has
singularities only where the denominator
''z''² + 1 is zero. Since
''z''² + 1 = (''z'' + ''i'')(''z'' − ''i''),
that happens only where ''z'' = ''i'' or ''z'' = −''i''.
Only one of those points is in the region bounded by this
contour.
Because ''f''(''z'') is
:{|
|-
| ||
ight),!
|-
| ||
-rac{e^{itz}}{2i(z+i)} , ,!
|}
the residue of
''f''(''z'') at ''z'' = ''i'' is
:$operatorname\{Res\}\_\{z=i\}f(z)=\{e^\{-t\}over\; 2i\}.$
According to the residue theorem, then, we have
:$int\_C\; f(z),dz=2pi\; icdotoperatorname\{Res\}\_\{z=i\}f(z)=2pi\; i\{e^\{-t\}\; over\; 2i\}=pi\; e^\{-t\}.$
The contour ''C'' may be split into a "straight"
part and a curved arc, so that
:$int\_\{mbox\{straight\}\}+int\_\{mbox\{arc\}\}=pi\; e^\{-t\},$
and thus
:$int\_\{-a\}^a\; =pi\; e^\{-t\}-int\_\{mbox\{arc\}\}.$
It can be shown that 'if ''t'' > 0 then'
:$int\_\{mbox\{arc\}\}\{e^\{itz\}\; over\; z^2+1\},dz$
ightarrow 0 mbox{as} a
ightarrowinfty.
Therefore 'if ''t'' > 0 then'
:$int\_\{-infty\}^infty\{e^\{itz\}\; over\; z^2+1\},dz=pi\; e^\{-t\}.$
A similar argument with an arc that winds around −''i''
rather than ''i'' shows that 'if ''t'' < 0 then'
:$int\_\{-infty\}^infty\{e^\{itz\}\; over\; z^2+1\},dz=pi\; e^t,$
and finally we have
:$int\_\{-infty\}^infty\{e^\{itz\}\; over\; z^2+1\},dz=pi\; e^\{-left|t$
ight|}.
(If ''t'' = 0 then the integral yields immediately to elementary calculus methods and its value is π.)

See also

★ Methods of contour integration

★ Morera's theorem

★ Nachbin's theorem

References

★ Complex Analysis, , Lars, Ahlfors, McGraw Hill, 1979,

External links

★ Residue theorem in MathWorld

★ Residue Theorem Module by John H. Mathews