hurwitzZeta

Hurwitz zeta function

Syntax

Z = hurwitzZeta(s,a)

Z = hurwitzZeta(n,s,a)

Description

Z = hurwitzZeta(s,a) evaluates the Hurwitz zeta function for the numeric or symbolic inputs s and a. The Hurwitz zeta function is defined only if s is not 1 and a is neither 0 nor a negative integer.

Z = hurwitzZeta(n,s,a) returns the nth derivative of hurwitzZeta(s,a) with respect to the variable s.

Examples

Numeric and Symbolic Inputs

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

Special Values

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

Differentiate Hurwitz Zeta Function

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 Hurwitz Zeta Function

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]);

Input Arguments

s — Input

number | array | symbolic number | symbolic variable | symbolic function | symbolic expression | symbolic array

Input, specified as a number, array, symbolic number, symbolic variable, symbolic function, symbolic expression, or symbolic array. The Hurwitz zeta function is defined only for values of s not equal to 1.

Data Types: single | double | sym | symfun
Complex Number Support: Yes

a — Input

number | array | symbolic number | symbolic variable | symbolic function | symbolic expression | symbolic array

Input, specified as a number, array, symbolic number, symbolic variable, symbolic function, symbolic expression, or symbolic array. The Hurwitz zeta function is defined only for values of a not equal to 0 or a negative integer.

Data Types: single | double | sym | symfun
Complex Number Support: Yes

n — Order of derivative

nonnegative integer

Order of derivative, specified as a nonnegative integer.

More About

Hurwitz Zeta Function

The Hurwitz zeta function is defined by the formula

$$\zeta \left(s,a\right)={\displaystyle \sum _{k=0}^{\infty }\frac{1}{{\left(k+a\right)}^s}}.$$

The summation series converges only when Re(s) > 1 and a is neither 0 nor a negative integer. Analytic continuation extends the definition of the function to the entire complex plane, except for a simple pole at s = 1.

Tips

• Floating-point evaluation of the Hurwitz zeta function can be slow for complex arguments or high-precision numbers. To increase the computational speed, you can reduce the floating-point precision by using the vpa and digits functions. For more information, see Increase Speed by Reducing Precision.

• The Hurwitz zeta function is related to other special functions. For example, it can be expressed in terms of the polylogarithm Li_s(z) and the gamma function Γ(z):

  $$\zeta \left(1-s,a\right)=\frac{\Gamma \left(s\right)}{{\left(2\pi \right)}^s}\left[e^{-i\pi s/2}{\text{Li}}_s\left(e^{2\pi ia}\right)+e^{i\pi s/2}{\text{Li}}_s\left(e^{-2\pi ia}\right)\right].$$

  Here, Re(s) > 0 and Im(a) > 0, or Re(s) > 1 and Im(a) = 0.

References

[1] Olver, F. W. J., A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., Chapter 25. Zeta and Related Functions, NIST Digital Library of Mathematical Functions, Release 1.0.20, Sept. 15, 2018.