JacobiDN

The Jacobi elliptic function $\operatorname{dn}(u,m)$ , one of the three primary Jacobi elliptic functions, in Wolfram/mpmath's $m=k^2$ parameter convention.

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

JacobiDN(u, m)the Jacobi elliptic delta amplitude, 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 — not the elliptic modulus k itself.
Numeric evaluation shares JacobiSN's descending Landen/arithmetic-geometric-mean kernel; for real u, dn comes from $\operatorname{dn}^2 = 1-m\operatorname{sn}^2$ (exact, no branch ambiguity there). For complex u, dn comes from DLMF 22.8's real addition formulas instead (real-u sn/cn/dn combined algebraically) — verified against mpmath to ~1e-13 relative across all four quadrants, large $\operatorname{Im}(u)$ , and close to dn's own pole structure.
A genuinely complex m stays unevaluated.
Exact at u = 0 (dn = 1, for any m), m = 0 (dn = 1), m = 1 (dn = 1/Cosh(u)), and the quarter period u = EllipticK(m) (dn = $\sqrt{1-m}$ ).
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 same AGM in BigDecimal for real $u$ and $0 < m < 1$ ; complex $u$ and other $m$ stay 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

23 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

In

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

Out

0.978\,240\,504\,174\,361\,3

Known value · mpmath 1.3.0 ellipfun

In

\operatorname{Series}(\mathrm{JacobiDN}(z, m), z, 0, 6)

Out

1+0mz-\frac{mz^2}{2}+\frac{1}{24}((m^2+4m)z^4)+\frac{1}{720}((-m^3-44m^2-16m)z^6)+O\left(z^8\right)

In

\operatorname{Series}(\mathrm{JacobiDN}(z, m), m, 0, 2)

Out

1+m×(\frac{\cos(2z)}{4}-\frac{1}{4})+(\frac{1}{8}(z\sin(2z))+\frac{\cos(4z)}{64}+\frac{\cos(2z)}{16}-\frac{5}{64})m^2+O\left(m^3\right)

Scope #

m greater than 1, via the reciprocal-modulus transformation

In

N(\mathrm{JacobiDN}(0.3, 1.5))

Out

0.935\,161\,917\,755\,299\,8

Known value · mpmath 1.3.0 ellipfun

m less than 0, via the imaginary-modulus transformation

In

N(\mathrm{JacobiDN}(0.3, -2))

Out

1.088\,512\,946\,606\,272\,4

Known value · mpmath 1.3.0 ellipfun

A complex argument u

In

N(\mathrm{JacobiDN}(0.3+0.4i, 0.5))

Out

1.012\,262\,082\,212\,854-0.062\,849\,685\,024\,787\,7i

Known value · mpmath 1.3.0 ellipfun

At $m=1$ , dn is sech, giving 2 at $u=-i\pi/3$

In

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

Out

\bigl\lbrack\bigl\lbrack2, 0\bigr\rbrack, \bigl\lbrack0, 2\bigr\rbrack\bigr\rbrack

In

\bigl\lbrack\mathrm{JacobiDN}(z, 0), \mathrm{JacobiDN}(z, 1)\bigr\rbrack

Out

\bigl\lbrack1, \frac{1}{\cosh(z)}\bigr\rbrack

In

N(\mathrm{JacobiDN}(3, \frac{1}{3}), 50)

Out

0.966\,263\,052\,100\,243\,189\,881\,465\,512\,089\,893\,511\,331\,758\,216\,268\,61

In

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

Out

\bigl\lbrack1, \mathrm{JacobiDN}(\frac{\mathrm{EllipticK}(m)}{2}, m), \sqrt{1-m}\bigr\rbrack

In

\operatorname{Series}(\mathrm{JacobiDN}(x, \frac{1}{3}), x, 0, 7)

Out

1-\frac{x^2}{6}+\frac{13x^4}{216}-\frac{277x^6}{19\,440}+O\left(x^8\right)

In

\operatorname{Series}(\mathrm{JacobiDN}(1, m), m, 0, 3)

Out

1+m×(\frac{\cos(2)}{4}-\frac{1}{4})+(-\frac{5}{64}+\frac{\cos(4)}{64}+\frac{\cos(2)}{16}+\frac{\sin(2)}{8})m^2+(-\frac{11}{256}-\frac{\cos(2)}{1\,024}+\frac{\cos(6)}{1\,024}+\frac{3\cos(4)}{256}+\frac{\sin(4)}{64}+\frac{9\sin(2)}{128})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{JacobiDN}(u, m)}{\mathrm{d}m}

Out

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

Properties #

u = 0 is 1, for any m

In

\mathrm{JacobiDN}(0, m)

Out

1

m = 0 is 1, for any u

In

\mathrm{JacobiDN}(u, 0)

Out

1

The quarter period u = EllipticK(m) is √(1−m)

In

\mathrm{JacobiDN}(\mathrm{EllipticK}(m), m)

Out

\sqrt{1-m}

dn(0) is 1

In

\mathrm{JacobiDN}(0, 1)

Out

1

Known value · Wolfram MathematicalFunctionData, JacobiDN

At $m=1$ , the pole sits at $i\pi/2$

In

\mathrm{JacobiDN}(\frac{1}{2}i π, 1)

Out

\tilde∞

Known value · Wolfram MathematicalFunctionData, JacobiDN

At $m=0$ dn is constant, even at imaginary multiples

In

\mathrm{JacobiDN}(3i\mathrm{EllipticK}(1), 0)

Out

1

Known value · Wolfram MathematicalFunctionData, JacobiDN

In

N((\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{2}, \frac{1}{2}))

Out

1.336\,269\,969\,158\,678\,193\cdot10^{-15}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N((\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{2}, \frac{1}{2}))

Out

2.883\,997\,392\,314\,445\,76\cdot10^{-16}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N(\mathrm{JacobiSN}(\frac{1}{2}, \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}+\mathrm{EllipticK}(\frac{1}{2}), \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiSN}(\frac{1}{2}+\mathrm{EllipticK}(\frac{1}{2}), \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}+\mathrm{EllipticK}(\frac{1}{2}), \frac{1}{2}))

Out

-1.125\,627\,516\,238\,416\,006\,225\,157\,470\,086\,064\cdot10^{-16}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N(\mathrm{JacobiDN}(\frac{2^{1/2}}{4}i, -1))

Out

0.935\,622\,687\,124\,812\,2

Known value · Wolfram MathematicalFunctionData, JacobiDN, Landen's transformation

In

N(\mathrm{JacobiDN}(\frac{1}{2}(1+\frac{2^{1/2}}{2}), \frac{(1-2^{1/2}/2)^2}{(1+2^{1/2}/2)^2}))

Out

0.991\,644\,164\,045\,051

Known value · Wolfram MathematicalFunctionData, JacobiDN, Gauss' transformation

In

N((\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{2}, \frac{1}{2}))

Out

1.336\,269\,969\,158\,678\,193\cdot10^{-15}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N((\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})+(\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{2}, \frac{1}{2}))

Out

2.883\,997\,392\,314\,445\,76\cdot10^{-16}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N(\sum_{k=0}^2(\mathrm{JacobiDN}(\frac{1}{3}(4(k-1)\mathrm{EllipticK}(1/2))+\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{3}(4(k+1)\mathrm{EllipticK}(1/2))+\frac{1}{2}, \frac{1}{2}))\mathrm{JacobiCN}(\frac{1}{3}(4k\mathrm{EllipticK}(1/2))+\frac{1}{2}, \frac{1}{2}))

Out

1.344\,615\,548\,852\,815\,796\cdot10^{-15}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N(\sum_{k=0}^2(\mathrm{JacobiDN}(\frac{1}{3}(4(k-1)\mathrm{EllipticK}(1/2))+\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{3}(4(k+1)\mathrm{EllipticK}(1/2))+\frac{1}{2}, \frac{1}{2}))\mathrm{JacobiSN}(\frac{1}{3}(4k\mathrm{EllipticK}(1/2))+\frac{1}{2}, \frac{1}{2}))

Out

3.880\,508\,872\,407\,314\,46\cdot10^{-16}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N(\mathrm{JacobiSN}(\frac{1}{2}, \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}+\mathrm{EllipticK}(\frac{1}{2}), \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}, \frac{1}{2})+\mathrm{JacobiSN}(\frac{1}{2}+\mathrm{EllipticK}(\frac{1}{2}), \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}+\mathrm{EllipticK}(\frac{1}{2}), \frac{1}{2}))

Out

-1.125\,627\,516\,238\,416\,006\,225\,157\,470\,086\,064\cdot10^{-16}

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

N(\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}, \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})+\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(4\mathrm{EllipticK}(1/2)), \frac{1}{2})\mathrm{JacobiDN}(\frac{1}{2}+\frac{1}{3}(8\mathrm{EllipticK}(1/2)), \frac{1}{2})\mathrm{JacobiCN}(\frac{1}{2}, \frac{1}{2}))

Out

-0.022\,694\,393\,112\,449\,704\,445\,776\,498\,109\,560\,672\,636\,671\,598\,743\,2

Known value · Wolfram MathematicalFunctionData, JacobiDN, FunctionalEquations

In

\mathrm{JacobiDN}(i\mathrm{EllipticK}(1), 0)

Out

1

Known value · Wolfram MathematicalFunctionData, JacobiDN

Possible issues #

Symbolic arguments are left unevaluated rather than guessed at

In

\mathrm{JacobiDN}(u, m)

Out

\mathrm{JacobiDN}(u, m)

About

Examples	55
Checked against	Wolfram Language 54 agree1 differ
Crosswalk	Wolfram Language

See also: JacobiSN, JacobiCN, JacobiAmplitude, EllipticK