jacobiZeta

Jacobi zeta function

Syntax

jacobiZeta(u,m)

Description

jacobiZeta(u,m) returns the Jacobi Zeta Function of u and m. If u or m is an array, then jacobiZeta acts element-wise.

Examples

Calculate Jacobi Zeta Function for Numeric Inputs

jacobiZeta(2,1)

ans =
    0.9640

Call jacobiZeta on array inputs. jacobiZeta acts element-wise when u or m is an array.

jacobiZeta([2 1 -3],[1 2 3])

ans =
   0.9640 + 0.0000i   0.5890 - 0.4569i  -2.3239 + 1.9847i

Calculate Jacobi Zeta Function for Symbolic Numbers

Convert numeric input to symbolic form using sym, and find the Jacobi zeta function. For symbolic input where u = 0 or m = 0 or 1, jacobiZeta returns exact symbolic output.

jacobiZeta(sym(2),sym(1))

ans =
tanh(2)

Show that for other values of u or m, jacobiZeta returns an unevaluated function call.

jacobiZeta(sym(2),sym(3))

ans =
jacobiZeta(2, 3)

Find Jacobi Zeta Function for Symbolic Variables or Expressions

For symbolic variables or expressions, jacobiZeta returns the unevaluated function call.

syms x y
f = jacobiZeta(x,y)

f =
jacobiZeta(x, y)

Substitute values for the variables by using subs, and convert values to double by using double.

f = subs(f, [x y], [3 5])

f =
jacobiZeta(3, 5)

fVal = double(f)

fVal =
   4.0986 - 3.0018i

Calculate f to arbitrary precision using vpa.

fVal = vpa(f)

fVal =
4.0986033838332279126523721581432 - 3.0017792319714320747021938869936i

Plot Jacobi Zeta Function

Plot real and imaginary values of the Jacobi zeta function using fcontour. Set u on the x-axis and m on the y-axis by using the symbolic function f with the variable order (u,m). Fill plot contours by setting Fill to on.

syms f(u,m)
f(u,m) = jacobiZeta(u,m);

subplot(2,2,1)
fcontour(real(f),'Fill','on')
title('Real Values of Jacobi Zeta')
xlabel('u')
ylabel('m')

subplot(2,2,2)
fcontour(imag(f),'Fill','on')
title('Imaginary Values of Jacobi Zeta')
xlabel('u')
ylabel('m')

Input Arguments

u — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or multidimensional array, or a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

m — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic variable | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or multidimensional array, or a symbolic number, variable, vector, matrix, multidimensional array, function, or expression.

More About

Jacobi Zeta Function

The Jacobi zeta function Z(u,m) is defined as

$$Z\left(u,m\right)=\frac{2\pi }{K(m)}\left({\displaystyle \sum _{i=1}^{\infty }\frac{q{(m)}^i}{1-q{(m)}^{2i}}}\sin \left(\frac{2\pi }{K(m)}iu\right)\right).$$

K(m) is the complete elliptic integral of the first kind, implemented as ellipticK. q(m) is the elliptic nome, implemented as ellipticNome.