hurwitzZeta

Evaluate the Hurwitz zeta function with numeric input arguments.

Z = hurwitzZeta(0,1)

Z = 
-0.5000

Compute the symbolic output of hurwitzZeta by converting the inputs to symbolic numbers using sym.

symZ = hurwitzZeta(sym([0 2]),1)

symZ = 
$$\left(\begin{array}{cc}-\frac{1}{2}& \frac{{\pi }^2}{6}\end{array}\right)$$

Use the vpa function to approximate symbolic results with the default 32 digits of precision.

valZ = vpa(symZ)

valZ = $$\left(\begin{array}{cc}-0.5& 1.644934066848226436472415166646\end{array}\right)$$

For certain parameter values, symbolic evaluation of the Hurwitz zeta function returns special values that are related to other symbolic functions.

For a = 1, the Hurwitz zeta function returns the Riemann zeta function zeta.

syms s a;
Z = hurwitzZeta(s,1)

Z = $$\zeta \text{<a href="matlab:doc symbolic/zeta">zeta</a>}\left(s\right)$$

For s = 2, the Hurwitz zeta function returns the first derivative of the digamma function psi.

Z = hurwitzZeta(2,a)

Z = $${\psi \text{<a href="matlab:doc symbolic/psi">psi</a>}}^{\prime }\left(a\right)$$

For nonpositive integers s, the Hurwitz zeta function returns polynomials in terms of a.

Z = hurwitzZeta(0,a)

Z = 
$$\frac{1}{2}-a$$

Z = hurwitzZeta(-1,a)

Z = 
$$-\frac{a^2}{2}+\frac{a}{2}-\frac{1}{12}$$

Z = hurwitzZeta(-2,a)

Z = 
$$-\frac{a^3}{3}+\frac{a^2}{2}-\frac{a}{6}$$

Find the first derivative of the Hurwitz zeta function with respect to the variable s.

syms s a
Z = hurwitzZeta(1,s,a)

Z = $${\zeta \text{<a href="matlab:doc symbolic/hurwitzZeta">hurwitzZeta</a>}}^{\prime }(s,a)$$

Evaluate the first derivative at s = 0 and a = 1 by using the subs function.

symZ = subs(Z,[s a],[0 1])

symZ = 
$$-\frac{\log \left(2\right)}{2}-\frac{\log \left(\pi \right)}{2}$$

Use the diff function to find the first derivative of the Hurwitz zeta function with respect to a.

Z = diff(hurwitzZeta(s,a),a)

Z = $$-s\hspace{0.17em}\zeta \text{<a href="matlab:doc symbolic/hurwitzZeta">hurwitzZeta</a>}(s+1,a)$$

Plot the Hurwitz zeta function for s within the interval [-20 10], given a = 0.7.

fplot(@(s) hurwitzZeta(s,0.7),[-20 10])
axis([-20 10 -40 35]);