Perturbative Methods for Modelling (M2)

Example 2, p. 38, Reference Book

Figure 2.1, p. 39, Reference Book

Exercice 2.6, p. 54, Reference Book

The Gamma function is defined by $\fs2 \Gamma(z) = \int_0^\infty e^{-t}\, t^{z-1}\, dt$ for $\fs2 {\rm Re}(z) > 0$.

• Show that $\fs2 \Gamma(1) = \Gamma(2) = 1$
• Show that $\fs2 \Gamma(1/2) = \sqrt\pi$
• Show that $\fs2 \Gamma(z+1) = z\, \Gamma(z)$

The Gamma function is extended to the complex plan with this last relation.

• Show that $\fs2 z=0, -1, -2,...$ are simple poles
• Compute the residues of $\fs2 \Gamma(z)$ for these poles