Editing Contour integration (section)

===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.