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 jacobiZeta(u,m) is defined as

$$Z\left(u,m\right)=E(\phi ,m)-\frac{E(m)}{K(m)}F(\phi ,m).$$

The definitions of the terms in above equation are:

• E(φ | m) and E(m) are the incomplete and complete elliptic integrals of the second kind, respectively, implemented as ellipticE.

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

• F(φ | m) is the incomplete elliptic integral of the first kind, implemented as ellipticF.

• am(u, m) is the Jacobi's amplitude function, implemented as jacobiAM.

The argument u is related to φ by the relations u = F(φ | m) and am(u, m) = φ, where am(u, m) is the Jacobi's amplitude function.

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, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., Chapter 22. Jacobian Elliptic Functions, NIST Digital Library of Mathematical Functions, Release 1.0.26 of 2020-03-15.