Editing Contour integration (section)

== Integral representation ==
{{expand section|date=November 2013}}
An '''integral representation''' of a function is an expression of the function involving a contour integral. Various integral representations are known for many [[special function]]s. Integral representations can be important for theoretical reasons, e.g. giving [[analytic continuation]] or [[functional equation]]s, or sometimes for [[numerical evaluation]]s.

[[Image:Hankel contour.png|right|thumb|150px|Hankel's contour]]

For example, the original definition of the [[Riemann zeta function]] {{math|''ζ''(''s'')}} via a [[Dirichlet series]],
<math display=block>\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s},</math>

is valid only for {{math|Re(''s'') > 1}}. But
<math display=block>\zeta(s) = - \frac{\Gamma(1 - s)}{2 \pi i} \int_H\frac{(-t)^{s-1}}{e^t - 1} dt ,</math>

where the integration is done over the [[Hankel contour]] {{mvar|H}}, is valid for all complex s not equal to 1.