zeta - Maple Help

MTM

 zeta
 the Riemann Zeta function

 Calling Sequence zeta(z) zeta(n,z)

Parameters

 z - algebraic expression n - (optional) algebraic expression, understood to be a non-negative integer

Description

 • The zeta function is defined for Re(z)>1 by

$\mathrm{zeta}\left(z\right)={\sum }_{i=1}^{\mathrm{\infty }}\frac{1}{{i}^{z}}$

and is extended to the rest of the complex plane (except for the point z=1) by analytic continuation. The point z=1 is a simple pole.

 • The call zeta(n, z) gives the nth derivative of the zeta function,

$\mathrm{zeta}\left(n,z\right)=\frac{{{ⅆ}}^{n}}{{ⅆ}{z}^{n}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{zeta}\left(z\right)$

 • zeta(z) will evaluate by default only when the result is an exact value, or when the input z is a floating point number.  When z is a symbolic expression, it will remain in function form so that it can be manipulated symbolically by itself or as part of a larger expression.
 • If z is an array or matrix, the result is an element-wise mapping over z.

Examples

 > $\mathrm{with}\left(\mathrm{MTM}\right):$
 > $\mathrm{zeta}\left(1,\frac{1}{2}\right)$
 ${\mathrm{\zeta }}{}\left(\frac{{1}}{{2}}\right){}\left(\frac{{\mathrm{\gamma }}}{{2}}{+}\frac{{\mathrm{ln}}{}\left({8}{}{\mathrm{\pi }}\right)}{{2}}{+}\frac{{\mathrm{\pi }}}{{4}}\right)$ (1)