Jacobi theta 1

Let $q \in \mathbb{C}$ with $|q|<1$. The Jacobi $\vartheta_1$ function is defined by $$\vartheta_1(z,q)=2q^{\frac{1}{4}} \displaystyle\sum_{k=0}^{\infty} (-1)^k q^{k(k+1)} \sin(2k+1)z,$$ where $\sin$ denotes the sine function.

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  Graph of $\vartheta_1(z,\frac{1}{2})$.

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  Domain coloring of $\vartheta_1 \left(z,\frac{1}{2} \right)$.

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
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 quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines

See also

Jacobi theta 2
Jacobi theta 3
Jacobi theta 4

References

• 1960: Earl David Rainville: Special Functions ... (previous) ... (next): $164. (1)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $16.27.1$