jtheta_ab {jacobi}

Jacobi theta function with characteristics

Description

Evaluates the Jacobi theta function with characteristics.

Usage

jtheta_ab(a, b, z, tau = NULL, q = NULL)

Arguments

a, b

the characteristics, two complex numbers

z

complex number, vector, or matrix

tau

lattice parameter, a complex number with strictly positive imaginary part; the two complex numbers tau and q are related by q = exp(1i*pi*tau), and only one of them must be supplied

q

the nome, a complex number whose modulus is strictly less than one, but not zero

Details

The Jacobi theta function with characteristics generalizes the four Jacobi theta functions. It is denoted by 𝜃[a,b](z|τ). One gets the four Jacobi theta functions when a and b take the values 0 or 0.5:

if a=b=0.5

then one gets -𝜗₁(z|τ)

if a=0.5 and b=0

then one gets 𝜗₂(z|τ)

if a=b=0

then one gets 𝜗₃(z|τ)

if a=0 and b=0.5

then one gets 𝜗₄(z|τ)

Both 𝜃[a,b](z+π|τ) and 𝜃[a,b](z+π×τ|τ) are equal to 𝜃[a,b](z|τ) up to a factor - see the examples for the details.

Value

A complex number, vector or matrix, like z.

Note

Different conventions are used in the book cited as reference.

References

Hershel M. Farkas, Irwin Kra. Theta Constants, Riemann Surfaces and the Modular Group: An Introduction with Applications to Uniformization Theorems, Partition Identities and Combinatorial Number Theory. Graduate Studies in Mathematics, volume 37, 2001.

Examples

a   <- 2 + 0.3i
b   <- 1 - 0.6i
z   <- 0.1 + 0.4i
tau <- 0.2 + 0.3i
jab <- jtheta_ab(a, b, z, tau) 
# first property ####
jtheta_ab(a, b, z + pi, tau) # is equal to:
jab * exp(2i*pi*a)
# second property ####
jtheta_ab(a, b, z + pi*tau, tau) # is equal to:
jab * exp(-1i*(pi*tau + 2*z + 2*pi*b))