Transformation laws of the Jacobi theta function

For $t>0$, every derivative of $f_t(u)=e^{-\pi t{u}^2}$ is a polynomial times this Gaussian. Exponential domination (The exponential dominates every fixed nonnegative integer power at $+\mathrm{\infty }$) bounds each such derivative times every power of $u$, so $f_t$ is Schwartz in Schwartz space and its seminorms. Apply [F2] at the real shift $a=x$ to obtain ${\sum }_n{e}^{-\pi t(n+x{)}^2}=t^{-1\mathrm{/}2}{\sum }_m{e}^{-\pi m^2\mathrm{/}t}{e}^{2\pi imx}$. Multiplying by $\sqrt{t}$ and expanding $(n+x{)}^2$ gives $\mathrm{\Theta }(x\mid i\mathrm{/}t)=\sqrt{t}{e}^{-\pi t{x}^2}\mathrm{\Theta }(itx\mid it)$ for every real $x$. This is the asserted inversion identity for $\tau =it$, since $-1\mathrm{/}(it)=i\mathrm{/}t$, $s(it)=\sqrt{t}$ and $e^{\pi i(it)x^2}=e^{-\pi t{x}^2}$. Countable choice is used precisely through the two suppliers in [F2].

Fix real $x$. Both sides of the claimed inversion identity are holomorphic in $\tau \in \mathfrak{H}$. For the right-hand theta series, on any compact $K\subset \mathfrak{H}$ its terms have modulus at most $e^{-\pi y_0{n}^2+C\mathrm{\mid }n\mathrm{\mid }}$, where $y_0={\min }_K\mathrm{Im}\tau >0$ and $C$ bounds $2\pi \mathrm{\mid }x\mathrm{Im}\tau \mathrm{\mid }$; this summable Gaussian majorant proves holomorphy by [F3]. The other factors are holomorphic by [F4], including $s$, since $\tau \mathrm{/}i$ lies in the right half-plane. The left side is holomorphic by [F1] and composition with $-1\mathrm{/}\tau$. By 1.1 the two sides agree on the positive imaginary axis, whose points accumulate within $\mathfrak{H}$, so the identity theorem [F3] proves equality for all $\tau$. Now fix such $\tau$. The two sides are entire in $z$ by [F1, F4] and agree for real $z=x$, so a second identity-theorem application proves the inversion formula for every $z\in \mathbb{C}$.

Each term of the theta series is unchanged on replacing $\tau$ by $\tau +2$, since $e^{2\pi i{n}^2}=1$ for integral $n$; absolute convergence therefore proves the stated period-two identity. Setting $z=0$ in 2.1 gives $\theta (-1\mathrm{/}\tau )=s(\tau )\theta (\tau )$. Since $n^2$ and $n$ have the same parity, [F1, F4] give $\theta (1+u)=\mathrm{\Theta }(1\mathrm{/}2\mid u)$ for $u\in \mathfrak{H}$. Apply 2.1 with $z=1\mathrm{/}2$ and complete the square: $e^{\pi i\tau \mathrm{/}4}\mathrm{\Theta }(\tau \mathrm{/}2\mid \tau )={\sum }_n{e}^{\pi i(n+1\mathrm{/}2{)}^2\tau }$. This proves the shifted-constant formula.

Pair $n=j$ with $n=-j-1$, $j\ge 0$, in the absolutely convergent shifted series. Its value is $2e^{\pi i\tau \mathrm{/}4}(1+{\sum }_{j\ge 1}{e}^{\pi i\tau j(j+1)})$. If $y=\mathrm{Im}\tau$, then $j(j+1)\ge 2j$ gives $\mathrm{\mid }{\sum }_{j\ge 1}{e}^{\pi i\tau j(j+1)}\mathrm{\mid }\le e^{-2\pi y}\mathrm{/}(1-e^{-2\pi y})$. This proves the uniform asymptotic and its stated error term. The principal square root never vanishes on $\mathfrak{H}$; no alternative square-root sign, boundary point $\tau =0$, or half-weight multiplier convention is implicit in any formula.