91Ó°ÊÓ

Prove that, for positive integral n:

$\mathbf{Γ}\left(n+\frac{1}{2}\right)\mathbf{=}\frac{\mathbf{1}\mathbf{·}\mathbf{3}\mathbf{·}\mathbf{5}\mathbf{⋯}\mathbf{(}\mathbf{2}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}}{{\mathbf{2}}^{\mathbf{n}}}\sqrt{\mathbf{π}}\mathbf{=}\frac{\mathbf{\left(}\mathbf{2}\mathbf{n}\mathbf{\right)}\mathbf{!}}{{\mathbf{4}}^{\mathbf{n}}\mathbf{n}\mathbf{!}}\sqrt{\mathbf{π}}$

Replace x by ix in (9.1) and let t = iuto show that erf(ix) = ierfi(x), where erfi(x) is defined in (9.7).

Use (5.4) to show that

(a) $\mathbf{Γ}\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{-}\mathbf{n}\mathbf{)}\mathbf{Γ}\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{+}\mathbf{n}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{n}}\mathit{π}$if $\mathit{n}\mathbf{=}\mathit{a}$ positive integer

(b)$\mathbf{(}\mathit{z}\mathbf{!}\mathbf{)}\mathbf{(}\mathbf{-}\mathit{z}\mathbf{)}\mathbf{!}\mathbf{=}\mathit{Ï€}\mathit{z}\mathbf{/}\mathit{s}\mathit{i}\mathit{n}\mathit{Ï€}\mathit{z}$, where zis not necessarily an integer; see comment after equation (3.3).

Prove that

$\begin{array}{c}\frac{\mathbf{d}}{\mathbf{d}\mathbf{p}}\mathbf{Γ}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}\underset{\mathbf{0}}{\overset{\mathbf{∞}}{\mathbf{∫}}}{\mathbf{x}}^{\mathbf{p}\mathbf{-}\mathbf{1}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}\mathbf{l}\mathbf{n}\mathbf{x}\mathbf{d}\mathbf{x}\\ \frac{{\mathbf{d}}^{\mathbf{n}}}{\mathbf{d}{\mathbf{p}}^{\mathbf{n}}}\mathbf{Γ}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{∞}}{\mathbf{x}}^{\mathbf{p}\mathbf{-}\mathbf{1}}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}{\mathbf{\left(}\mathbf{l}\mathbf{n}\mathbf{x}\mathbf{\right)}}^{\mathbf{n}}\mathbf{d}\mathbf{x}\end{array}$

Assuming that x is real, show the following relation between the error function and the Fresnel integrals.

$\mathit{e}\mathit{r}\mathit{f}\mathbf{(}\frac{\mathbf{1}\mathbf{-}\mathbf{i}}{\sqrt{\mathbf{2}}}\mathbf{.}\mathbf{x}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{)}\sqrt{\frac{\mathbf{2}}{\mathbf{π}}}\underset{\mathbf{0}}{\overset{\mathbf{x}}{\mathbf{∫}}}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}{\mathbf{u}}^{\mathbf{2}}\mathbf{+}\mathbf{i}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}{\mathbf{u}}^{\mathbf{2}}\mathbf{)}\mathit{d}\mathit{u}$

In the Table of Laplace Transforms (end of Chapter 8, page 469), verify$\mathit{Γ}$the function results for L5 and L6. Also show that$\mathit{L}\mathbf{(}\mathbf{1}\mathbf{/}\sqrt{\mathbf{t}}\mathbf{)}\mathbf{=}\sqrt{\mathbf{π}\mathbf{/}\mathbf{p}}$.