Editing Theta function (section)

=== Further values ===
Many values of the theta function<ref>{{cite journal |last1=Yi |first1=Jinhee |title=Theta-function identities and the explicit formulas for theta-function and their applications |journal=Journal of Mathematical Analysis and Applications |date=15 April 2004 |volume=292 |issue=2 |pages=381–400 |doi=10.1016/j.jmaa.2003.12.009 |doi-access=free }}</ref> and especially of the shown phi function can be represented in terms of the gamma function:

:<math>\begin{array}{lll}
\varphi\left(\exp( -\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{7/8} \\
\varphi\left(\exp(-2\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{1/8}\Bigl(1+\sqrt{\sqrt{2}-1}\Bigr) \\
\varphi\left(\exp(-3\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{3/8}3^{-1/2}(\sqrt{3}+1)\sqrt{\tan(\tfrac{5}{24}\pi)} \\
\varphi\left(\exp(-4\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2}2^{-1/8}\Bigl(1+\sqrt[4]{2\sqrt{2}-2}\Bigr) \\
\varphi\left(\exp(-5\sqrt{2}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{9}{8}\right){\Gamma\left(\tfrac{5}{4}\right)}^{-1/2} \frac{1}{15}\,2^{3/8} \times \\
 && \times \biggl[\sqrt[3]{5}\,\sqrt{10+2\sqrt{5}}\biggl(\sqrt[3]{5+\sqrt{2}+3\sqrt{3}}+\sqrt[3]{5+\sqrt{2}-3\sqrt{3}}\,\biggr)-\bigl(2-\sqrt{2}\,\bigr)\sqrt{25-10\sqrt{5}}\,\biggr] \\
\varphi\left(\exp( -\sqrt{6}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{5}{24}\right){\Gamma\left(\tfrac{5}{12}\right)}^{-1/2}2^{-13/24}3^{-1/8}\sqrt{\sin(\tfrac{5}{12}\pi)} \\
\varphi\left(\exp(-\tfrac{1}{2}\sqrt{6}\,\pi)\right) &=& \pi^{-1/2}\Gamma\left(\tfrac{5}{24}\right){\Gamma\left(\tfrac{5}{12}\right)}^{-1/2}2^{5/24}3^{-1/8}\sin(\tfrac{5}{24}\pi)
\end{array}</math>