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==Example==

==Example==

Here is a simple example. Suppose we want to find the [[integral]] of <math>\log(x)/(1+x^2)</math> from 0 to <math>\infty</math>. We begin by choosing an [[analytic]] branch of the [[logarithm]], defined everywhere in <math>\mathbb{C}</math> except for the line consisting of negative purely imaginary numbers (that is, <math>\log(z)=\log|z|+i\arg(z)</math> where <math>\arg(z)</math> is specified to take values in <math>(-\pi/2,3\pi/2)</math>. Now, choose real numbers <math>r,R</math> where <math>0<r<1</math> and <math>R>1</math>. Then, let <math>\Gamma(r,R)</math> be the positively oriented [[contour]] consisting of the clockwise upper semicircular [[arc]] from <math>-r</math> to <math>r</math>, the directed [[line segment]] from <math>r</math> to <math>R</math>, the anticlockwise semicircular arc from <math>R</math> to <math>-R</math>, and the directed line segment from <math>-R</math> to <math>-r</math> (this looks like a rainbow in the complex plane).

Here is a simple example. Suppose we want to find the [[integral]] of <math>\log(x)/(1+x^2)</math> from 0 to <math>\infty</math>. We begin by choosing an [[analytic]] branch of the [[logarithm]], defined everywhere in <math>\mathbb{C}</math> except for the line consisting of negative purely imaginary numbers (that is, <math>\log(z)=\log|z|+i\arg(z)</math>) where <math>\arg(z)</math> is specified to take values in <math>(-\pi/2,3\pi/2)</math>. Now, choose real numbers <math>r,R</math> where <math>0<r<1</math> and <math>R>1</math>. Then, let <math>\Gamma(r,R)</math> be the positively oriented [[contour]] consisting of the clockwise upper semicircular [[arc]] from <math>-r</math> to <math>r</math>, the directed [[line segment]] from <math>r</math> to <math>R</math>, the anticlockwise semicircular arc from <math>R</math> to <math>-R</math>, and the directed line segment from <math>-R</math> to <math>-r</math> (this looks like a rainbow in the complex plane).

The reason we chose this contour is that it necessarily avoids the inevitable singularity of the logarithm at 0, and it includes inside of it, the pole of the function <math>f(z)=log(z)/(1+z^2)</math>. By Cauchy's residue theorem, we have <math>I:=\int_{\Gamma(r,R)}f(z)\,dz</math>= <math>2\pi i</math> times the residue of the [[pole]] at <math>z=i</math>. This is easily calculated, and it is equal to <math>\pi/4</math>. Thus, the integral evaluates to <math>\pi^2 i/2</math>.

The reason we chose this contour is that it necessarily avoids the inevitable singularity of the logarithm at 0, and it includes inside of it, the pole of the function <math>f(z)=log(z)/(1+z^2)</math>. By Cauchy's residue theorem, we have <math>I:=\int_{\Gamma(r,R)}f(z)\,dz</math>= <math>2\pi i</math> times the residue of the [[pole]] at <math>z=i</math>. This is easily calculated, and it is equal to <math>\pi/4</math>. Thus, the integral evaluates to <math>\pi^2 i/2</math>.