JacobiZN

Wolfram's Jacobi zeta function $Z(u,m) = E(\operatorname{am}(u,m),m) - \dfrac{E(m)}{K(m)}u$ , in Wolfram/mpmath's $m=k^2$ convention.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageJacobiZN!

JacobiZN(u, m)the Jacobi zeta function, argument u, parameter m.

Typed signatures

Type	Package	Applies
`(number, number) -> number`	`analytic`	always

Domain: Special functions

Details

$m = k^2$ , the same convention EllipticK / EllipticF use.
Composed from JacobiAmplitude's amplitude and the already-declared IncompleteEllipticE, EllipticE, EllipticK (DLMF 22.16.31) — not a separate numeric method.
Real m between 0 and 1 only, inheriting JacobiAmplitude's restriction (no verified amplitude transform for m outside that range).
Z(0,m) = 0, for any m in range; Z(u,0) = 0, for any u — am(u,0) = u makes E(am,0) = u exactly, canceling the E(0)/K(0)·u = u term.
Numeric only — a symbolic argument (outside the exact table above) stays unevaluated; a floating-point argument (or N()) evaluates directly.
N(x, d) past a double's digits runs the AGM amplitude in BigDecimal and takes $E$ and $K$ from the Carlson kernels, for real $u$ and $0 < m < 1$ ; anything else stays unevaluated there rather than print a double's digits as more.
Derivatives of every order in $u$ and in $m$ are closed forms in sn, cn, dn and the Jacobi epsilon $\mathcal{E}(u,m)=Z(u,m)+u\,E(m)/K(m)$ (DLMF 22.13): D, Series in $u$ at $0$ or at a symbolic point, and Series in $m$ at $m=0$ all give concrete coefficients. The $m$ -derivatives evaluate numerically for real $u$ and $m$ between 0 and 1.

Examples

2 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

In

N(\mathrm{JacobiZN}(0.3, 0.5))

Out

0.077\,076\,468\,477\,655\,76

Known value · Wolfram 15.0.0 JacobiZN

In

\operatorname{Series}(\mathrm{JacobiZN}(u, m), u, 0, 7)

Out

u×(1-\frac{\mathrm{EllipticE}(m)}{\mathrm{EllipticK}(m)})+0mu^2-\frac{mu^3}{3}+\frac{1}{15}((m^2+m)u^5)+\frac{1}{315}((-2m^3-13m^2-2m)u^7)+O\left(u^9\right)

In

\operatorname{Series}(\mathrm{JacobiZN}(u, m), m, 0, 2)

Out

\frac{1}{4}(m\sin(2u))+(-\frac{1}{8}(u\cos(2u))+\frac{\sin(4u)}{64}+\frac{\sin(2u)}{16})m^2+O\left(m^3\right)

Scope #

A parameter close to its m = 1 boundary

In

N(\mathrm{JacobiZN}(1.1, 0.9))

Out

0.354\,712\,200\,199\,871\,26

Known value · Wolfram 15.0.0 JacobiZN

At $m=1$ , the zeta function is $\tanh$ , giving $-i\sqrt3$

In

\mathrm{MatrixFunction}(()↦\mathrm{JacobiZN}((-1/3i)π, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)

Out

\bigl\lbrack\bigl\lbrack-\sqrt{3}i, 0\bigr\rbrack, \bigl\lbrack0, -\sqrt{3}i\bigr\rbrack\bigr\rbrack

Zeta is 0 at $m=0$ and $\tanh u$ at $m=1$

In

\bigl\lbrack\mathrm{JacobiZN}(u, 0), \mathrm{JacobiZN}(u, 1)\bigr\rbrack

Out

\bigl\lbrack0, \tanh(u)\bigr\rbrack

In

N(\mathrm{JacobiZN}(1, \frac{1}{2}), 50)

Out

0.143\,801\,986\,499\,132\,943\,480\,315\,826\,578\,906\,420\,121\,478\,723\,940\,84

In

\bigl\lbrack\mathrm{JacobiZN}(0, m), \mathrm{JacobiZN}(\frac{\mathrm{EllipticK}(m)}{2}, m), \mathrm{JacobiZN}(\mathrm{EllipticK}(m), m)\bigr\rbrack

Out

\bigl\lbrack0, \mathrm{JacobiZN}(\frac{\mathrm{EllipticK}(m)}{2}, m), \mathrm{JacobiZN}(\mathrm{EllipticK}(m), m)\bigr\rbrack

In

\operatorname{Series}(\mathrm{JacobiZN}(u, \frac{1}{3}), u, 0, 7)

Out

u×(1-\frac{\mathrm{EllipticE}(\frac{1}{3})}{\mathrm{EllipticK}(\frac{1}{3})})-\frac{u^3}{9}+\frac{4u^5}{135}-\frac{59u^7}{8\,505}+O\left(u^9\right)

In

\operatorname{Series}(\mathrm{JacobiZN}(2, m), m, 0, 3)

Out

\frac{m\sin(4)}{4}+(-\frac{\cos(4)}{4}+\frac{\sin(8)}{64}+\frac{\sin(4)}{16})m^2+(-\frac{9\cos(4)}{64}-\frac{95\sin(4)}{1\,024}-\frac{\cos(8)}{32}+\frac{\sin(12)}{1\,024}+\frac{3\sin(8)}{256})m^3+O\left(m^4\right)

The derivative in the parameter m, in closed form through the Jacobi zeta

In

\frac{\mathrm{d}\mathrm{JacobiZN}(u, m)}{\mathrm{d}m}

Out

\frac{u\mathrm{EllipticE}(m)^2}{2m×(1-m)\mathrm{EllipticK}(m)^2}-\frac{u\mathrm{EllipticE}(m)}{m\mathrm{EllipticK}(m)}-\frac{(\frac{u\mathrm{EllipticE}(m)}{\mathrm{EllipticK}(m)}+\mathrm{JacobiZN}(u, m))\mathrm{JacobiCN}(u, m)^2}{2(1-m)}+\frac{u}{2m}-\frac{1}{2}(u\mathrm{JacobiSN}(u, m)^2)+\frac{\mathrm{JacobiSN}(u, m)\mathrm{JacobiDN}(u, m)\mathrm{JacobiCN}(u, m)}{2(1-m)}

Properties #

u = 0 is 0, for any m

In

\mathrm{JacobiZN}(0, m)

Out

0

m = 0 is 0, for any u

In

\mathrm{JacobiZN}(u, 0)

Out

0

Possible issues #

m outside [0, 1] is declined — no verified amplitude transform there

In

N(\mathrm{JacobiZN}(0.3, -0.5))

Out

\mathrm{JacobiZN}(0.3, -0.5)

Symbolic arguments are left unevaluated rather than guessed at

In

\mathrm{JacobiZN}(u, m)

Out

\mathrm{JacobiZN}(u, m)

About

Examples	17
Checked against	Wolfram Language 15 agree2 differ
Crosswalk	Wolfram Language

See also: JacobiAmplitude, EllipticE, EllipticK