Editing Contour integration (section)

==Contour integrals==
The '''contour integral''' of a [[complex function]] <math>f:\C\to\C</math> is a generalization of the integral for real-valued functions. For [[continuous function]]s in the [[complex plane]], the contour integral can be defined in analogy to the [[line integral]] by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the [[partition of an interval]] and the [[Riemann integral]]. In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour.

===For continuous functions===
To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let <math>f:\R\to\C</math> be a complex-valued function of a real variable, <math>t</math>. The real and imaginary parts of <math>f</math> are often denoted as <math>u(t)</math> and <math>v(t)</math>, respectively, so that
<math display=block>f(t) = u(t) + iv(t).</math>
Then the integral of the complex-valued function <math>f</math> over the interval <math>[a,b]</math> is given by 
<math display=block>\begin{align}
\int_a^b f(t) \, dt &= \int_a^b \big( u(t) + i v(t) \big) \, dt \\
&= \int_a^b u(t) \, dt + i \int_a^b v(t) \, dt. 
\end{align}</math>

Now, to define the contour integral, let <math>f:\C\to\C</math> be a [[continuous function]] on the [[Methods of contour integration#Directed smooth curves|directed smooth curve]] <math>\gamma</math>. Let <math>z:[a,b]\to\C</math> be any parametrization of <math>\gamma</math> that is consistent with its order (direction). Then the integral along <math>\gamma</math> is denoted
<math display=block>\int_\gamma f(z)\, dz\, </math>
and is given by<ref name=Saff/>
<math display="block">\int_\gamma f(z) \, dz := \int_a^b f\big(z(t)\big) z'(t) \, dt.</math>

This definition is well defined. That is, the result is independent of the parametrization chosen.<ref name="Saff" /> In the case where the real integral on the right side does not exist the integral along <math>\gamma</math> is said not to exist.

===As a generalization of the Riemann integral===
The generalization of the [[Riemann integral]] to functions of a complex variable is done in complete analogy to its definition for functions from the real numbers. The partition of a directed smooth curve <math>\gamma</math> is defined as a finite, ordered set of points on <math>\gamma</math>. The integral over the curve is the limit of finite sums of function values, taken at the points on the partition, in the limit that the maximum distance between any two successive points on the partition (in the two-dimensional complex plane), also known as the mesh, goes to zero.