jacobiZeta
Jacobi zeta function
Syntax
Description
jacobiZeta(
returns the Jacobi Zeta Function of
u
,m
)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
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.
Version History
Introduced in R2017b
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