EllipticTheta

The Jacobi theta function $\vartheta_a(u,q)$, $a=1,2,3,4$ (DLMF 20.2), Wolfram's nome convention $|q|<1$.

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

EllipticTheta(a, u, q)the a-th Jacobi theta function (a = 1, 2, 3, or 4), argument u, nome q.

Typed signatures

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

Domain: Special functions

Details

• $q$ is the nome, $|q| < 1$ — the same convention mpmath's jtheta(a, u, q) uses, not the "half-period ratio" $\tau$ some other references parameterize by.
• Direct q-series (DLMF 20.2.1-20.2.4), term-summed until three consecutive terms fall under $10^{-15}$ or 20000 terms are reached.
• Declines when $|q| \ge 1$ (the series does not converge there at all), or when $|q|$ is close enough to 1 that convergence isn't reached within that term cap — no modular (Jacobi imaginary) transform is implemented to extend the series further.
• $a$ must be a concrete integer 1, 2, 3, or 4; any other value (including a symbolic a) stays unevaluated.
• Verified against mpmath's jtheta and a Wolfram kernel at complex u, complex q, AND real negative q.
• Numeric only — a symbolic u or q stays unevaluated; a floating-point argument (or N()) evaluates directly.

Examples

22 more kept as data, checked against the oracles

Basic #

A floating-point argument evaluates directly to a decimal

live ↗

In$N(\mathrm{EllipticTheta}(3, 0.3, 0.2))$Out$1.331\,293\,558\,113\,586\,5$

Known value · mpmath 1.3.0 jtheta

Scope #

Each of the four theta functions (a = 1)

live ↗

In$N(\mathrm{EllipticTheta}(1, 0.3, 0.2))$Out$0.353\,430\,543\,747\,624\,5$

Known value · mpmath 1.3.0 jtheta

A complex argument u

live ↗

In$N(\mathrm{EllipticTheta}(3, 0.3+0.1i, 0.2))$Out$1.338\,012\,245\,244\,520\,2-0.046\,698\,857\,386\,200\,96i$

Known value · mpmath 1.3.0 jtheta

A negative real nome q

live ↗

In$N(\mathrm{EllipticTheta}(3, 1, -0.3))$Out$1.239\,061\,275\,954\,273$

Known value · mpmath 1.3.0 jtheta

A complex nome q, together with a complex u

live ↗

In$N(\mathrm{EllipticTheta}(2, 0.5+0.5i, 0.1+0.2i))$Out$1.482\,111\,803\,583\,081\,8+0.163\,554\,249\,597\,655\,97i$

Known value · mpmath 1.3.0 jtheta

At nome $q=0$ only the constant term survives: $\vartheta_3(u,0)=1$

live ↗

In$\mathrm{EllipticTheta}(3, 0.5, 0)$Out$1$

Properties #

theta1(0,q) is 0 for any q — an odd function of u

live ↗

In$N(\mathrm{EllipticTheta}(1, 0, 0.5))$Out$0.0$

Known value · mpmath 1.3.0 jtheta

Possible issues #

|q| ≥ 1 is declined — the series diverges there

live ↗

In$N(\mathrm{EllipticTheta}(3, 0.3, 1))$Out$\mathrm{EllipticTheta}(3, 0.3, 1)$

Symbolic arguments are left unevaluated rather than guessed at

live ↗

In$\mathrm{EllipticTheta}(3, u, q)$Out$\mathrm{EllipticTheta}(3, u, q)$

u = 10^30 spans far more than 10^6 periods of π: u this far out has no argument reduction against its real period, so the guard in exceedsDoublePrecision's sibling (periodsExceedDouble, ce-patches/support/precise.ts) declines rather than return a wrong value.

live ↗

In$N(\mathrm{EllipticTheta}(1, 10^{30}, \frac{1}{2}))$Out$\mathrm{EllipticTheta}(1, 1.0\cdot10^{30}, 0.5)$

The true value (mpmath, dps=30) — still not computed here. Argument reduction at bignum precision (reduce u mod π before collapsing to a double) is deferred; this example only pins the gap, not a fix.

live ↗

Known value · mpmath 1.3.0 jtheta

About

Examples	33
Checked against	Wolfram Language 31 agree1 differ
Crosswalk	Wolfram Language

See also: EllipticThetaPrime, JacobiSN