We have the following integral representation for the full two-parameter form of Ramanujan's theta function: : \begin{align} f(a,b) = 1 + \int_0^{\infty} \frac{2a e^{-\frac12 t^2}}{\sqrt{2\pi}}\left[ \frac{1 - a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{ a^3 b - 2a \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1} \right] dt + \\ \int_0^{\infty} \frac{2b e^{-\frac12 t^2}}{\sqrt{2\pi}}\left[ \frac{1 - b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right)}{ a b^3 - 2b \sqrt{ab} \cosh\left(\sqrt{\log ab} \,t\right) + 1} \right] dt \end{align} The special cases of Ramanujan's theta functions given by and also have the following integral representations: : \begin{align} \varphi(q) & = 1 + \int_0^{\infty} \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[\frac{4q \left(1-q^2 \cosh\left( \sqrt{2 \log q} \,t\right)\right)}{q^4-2 q^2 \cosh\left(\sqrt{2 \log q} \,t\right) + 1} \right] dt \\[6pt] \psi(q) & = \int_0^{\infty} \frac{2 e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[\frac{1-\sqrt{q} \cosh\left(\sqrt{\log q} \,t\right)}{q-2 \sqrt{q} \cosh\left(\sqrt{\log q} \,t\right) + 1} \right] dt \end{align} This leads to several special case integrals for constants defined by these functions when (cf.
theta function explicit values). In particular, we have that : \begin{align} \varphi\left(e^{-k\pi}\right) & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{k\pi} \left(e^{2k\pi} - \cos\left(\sqrt{2\pi k} \,t\right) \right)}{e^{4k\pi} - 2 e^{2k\pi} \cos\left(\sqrt{2\pi k} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^\pi \left(e^{2\pi} - \cos\left(\sqrt{2\pi} \,t\right) \right)}{e^{4\pi} - 2 e^{2\pi} \cos\left(\sqrt{2\pi} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{2\pi} \left(e^{4\pi} - \cos\left(2 \sqrt{\pi} \,t\right) \right)}{e^{8\pi} - 2 e^{4\pi} \cos\left(2 \sqrt{\pi} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt{1 + \sqrt{3}}}{2^\frac14 3^\frac38} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{3\pi} \left(e^{6\pi} - \cos\left(\sqrt{6 \pi} \,t\right) \right)}{e^{12\pi} - 2 e^{6\pi} \cos\left(\sqrt{6 \pi} \,t\right) + 1} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt{5 + 2 \sqrt{5}}}{5^\frac34} & = 1 + \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{4 e^{5\pi} \left(e^{10\pi} - \cos\left(\sqrt{10 \pi} \,t\right) \right)}{e^{20\pi} - 2 e^{10\pi} \cos\left(\sqrt{10 \pi} \,t\right) + 1} \right] dt \end{align} and that : \begin{align} \psi\left(e^{-k\pi}\right) & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{k \pi} \,t\right) - e^\frac{k\pi}{2}}{ \cos\left(\sqrt{k \pi} \,t\right) - \cosh\frac{k\pi}{2}} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{e^\frac{\pi}{8}}{2^\frac58} & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{\pi} \,t\right) - e^\frac{\pi}{2}}{ \cos\left(\sqrt{\pi} \,t\right) - \cosh\frac{\pi}{2}} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{e^\frac{\pi}{4}}{2^\frac54} & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{2 \pi} \,t\right) - e^\pi}{ \cos\left(\sqrt{2 \pi} \,t\right) - \cosh \pi} \right] dt \\[6pt] \frac{\pi^\frac14}{\Gamma\left(\frac34\right)} \cdot \frac{\sqrt[4]{1 + \sqrt{2}} \, e^\frac{\pi}{16}}{2^\frac{7}{16}} & = \int_0^\infty \frac{e^{-\frac12 t^2}}{\sqrt{2\pi}} \left[ \frac{\cos\left(\sqrt{\frac{\pi}{2}} \,t\right) - e^\frac{\pi}{4}}{ \cos\left(\sqrt{\frac{\pi}{2}} \,t\right) - \cosh\frac{\pi}{4}} \right] dt \end{align} ==Application in string theory==