Editing Hankel contour (section)

== Applications ==

=== The Hankel contour and the Gamma function ===
The Hankel contour is helpful in expressing and solving the Gamma function in the complex ''t''-plane. The Gamma function can be defined for any [[Complex number|complex value]] in the plane if we evaluate the integral along the Hankel contour. The Hankel contour is especially useful for expressing the Gamma function for any complex value because the end points of the contour vanish, and thus allows the fundamental property of the Gamma function to be satisfied, which states <math>\Gamma(z+1)=z\Gamma(z)</math>.<ref name=":1" />

==== Derivation of the contour integral expression of the Gamma function ====
The Hankel contour can be used to help derive an expression for the Gamma function,<ref name=":1" />
based on the fundamental property <math>\Gamma(z+1) = z \Gamma(z)</math>.
Assume an [[ansatz]] of the form <math>\Gamma(z)=\int_C f(t)t^{z-1} dt</math>, where <math>C</math> is the Hankel contour.

Inserting this ansatz into the fundamental property and integrating by parts on the right-hand side, one obtains
<math display="block">\int_C f(t)t^z dt = [t^z f(t)] - \int_C t^z f'(t)dt.</math>

Thus, assuming <math>f(t)</math> decays sufficiently quickly such that <math>t^z f(t)</math> vanishes at the endpoints of the Hankel contour,
<math display="block">\int_C t^z (f(t) + f'(t)) dt = 0 \implies f(t)+f'(t)=0.</math>

The solution to this [[differential equation]] is
<math>f(t)=Ae^{-t}.</math>
While <math>A</math> is a constant with respect to <math>t</math>, <math>A</math> may nonetheless be a function of <math>z</math>. 
Substituting <math>f(t)</math> into the original integral then gives
<math display="block">\Gamma(z)=A(z)\int_C e^{-t} (-t)^{z-1}dt,</math>
where the minus sign in <math>(-t)^{z-1}</math> is accounted for by absorbing a factor <math>(-1)^{z-1}</math> into the definition of <math>A(z)</math>.

By integrating along the Hankel contour, the contour integral expression of the Gamma function becomes{{clarify|date=October 2024}} <math>\Gamma(z)=\frac{i}{2\sin{\pi z}}\int_C e^{-t}(-t)^{z-1}dt</math>.<ref name=":1" />

<b>Proof</b> :
As stated above, the integral I<sub>c</sub> over the entire contour C is the sum of three integrals:
* The one on the semi-axis ]+∞, M] of imaginary part +iδ which will be referred as <math>I_{\delta}^{+}</math>
* That on the partial circle between M and N that will be referred as Iε
* The one on the semi-axis [N, +∞[ of imaginary part -iδ which will be referred as <math>I_{\delta}^{-}</math>
Iε can be discarded as it tends to 0 when ε tends to 0.

For <math>I_{\delta}^{+}</math> and <math>I_{\delta}^{-}</math>, the expression <math>(-t)^{z-1}</math> will be put in the form <math>e^{(z-1)log(-t)}</math>.

Any complex c can be written as ρ cos(θ) + i sin(θ), or ρ <math>e^{i\theta}</math> with positive or zero real ρ and real θ. If one requires that θ be between -π (not included) and +π, θ is unique if c is not zero. θ is named the argument of the complex c and ρ is its modulus (unique in all cases).
The complex logarithm of c is defined as being equal to log(ρ) + iθ, log(ρ) being the usual real logarithm and θ belonging to ]-π, π].