Jacobi theta 4

Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_4$ function is defined by $$\vartheta_4(z,q)=1+2\displaystyle\sum_{k=1}^{\infty} (-1)^k q^{k^2} \cos(2kz),$$ where $\cos$ denotes the cosine function.

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  Graph of $\vartheta_4(z,\frac{1}{2})$.

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  Domain coloring of $\vartheta_4 \left(z,\frac{1}{2} \right)$.

Properties

Squares of theta relation for Jacobi theta 1 and Jacobi theta 4
Squares of theta relation for Jacobi theta 2 and Jacobi theta 4
Squares of theta relation for Jacobi theta 3 and Jacobi theta 4
Squares of theta relation for Jacobi theta 4 and Jacobi theta 4
Sum of fourth powers of Jacobi theta 2 and Jacobi theta 4 equals fourth power of Jacobi theta 3
Derivative of Jacobi theta 1 at 0
Logarithm of a quotient of Jacobi theta 4 equals a sum of sines

See also

Jacobi theta 1
Jacobi theta 2
Jacobi theta 3

References

• 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $164. (4)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.27.4$