I am trying to determine in what way to approach finding a connection between Dedekind's Eta Function, defined as $$\eta(\tau)=q^\frac{1}{24}\prod_{n=1}^\infty(1-q^n)$$
where $q=e^{2\pi i \tau}$ is referred to as the nome.
and the Gamma Function $$\Gamma(s)=\int_{0}^\infty x^{s-1}e^{-x}dx$$ More specifically I would like to understand through what methods these identities are derived: $$\eta(i)=\Gamma(\frac{1}{4})\frac{\pi^{-3/4}}{2}$$
$$\eta(2i)=\Gamma(\frac{1}{4})2^{-11/8}\pi^{-3/4}$$
And in general what seems to be $$\eta(ki)=\Gamma(\frac{1}{4})\pi^{-3/4}C_{k}$$
for whole numbers $k$ and some constant $C_k$ Where $C_k$ looks to be algebraic for $k\in 1,2,3,4$.
I guess what I really want to know is why does this factor of $\Gamma(\frac{1}{4})\pi^{-3/4}$ come into play at imaginary integer values for the $\eta$ function?
I know there is a relationship between the $\eta$ and Jacobi Theta Functions that can be found using the Pentagonal Number Theorem or Jacobi's Triple Product Identity but I do not know how it fits into evaluation of $\eta(ki)$.
EDIT:My attempt at an answer: $$\int_{-\infty}^\infty e^{-x^{2p}} dx=\frac{\Gamma(\frac{1}{2p})}{p}$$ can be derived through substitution. $$\frac{\Gamma^2(\frac{1}{2p})}{p^2}=\int_\Bbb {R^2}\exp(-(x^{2p}+y^{2p})dxdy$$
Applying the coordinate transformation $x^{2p}+y^{2p}=r^{2p}$ with $x=r\frac{\cos(\phi)}{|\sin(\phi)|^{2p}+|cos(\phi)|^{2p}}$ and $y=r\frac{\sin(\phi)}{|\sin(\phi)|^{2p}+|cos(\phi)|^{2p}}$ I get$$\frac{\Gamma^2(\frac{1}{2p})}{p^2}=\int_{0}^\infty re^{-r^{2p}}dr\int_{0}^{2\pi}\frac{d\phi}{(\sin^{2p}(\phi)+\cos^{2p}(\phi))^{\frac{1}{p}}}$$
The integral over $r$ evaluates to $\frac{\Gamma(\frac{1}{p})}{2p}$
So$$\frac{2\Gamma^2(\frac{1}{2p})}{p\Gamma({\frac{1}{p})}}=\int_{0}^{2\pi}\frac{d\phi}{(\sin^{2p}(\phi)+\cos^{2p}(\phi))^{\frac{1}{p}}}$$
The integral is symmetric over $[0,\pi]$ and $[\pi, 2\pi]$ so we get $$\frac{\Gamma^2(\frac{1}{2p})}{p\Gamma({\frac{1}{p})}}=\int_{0}^{\pi}\frac{d\phi}{(\sin^{2p}(\phi)+\cos^{2p}(\phi))^{\frac{1}{p}}}$$ Plugging in $p=2$ yields $$\frac{\Gamma^2(\frac{1}{4})}{2\sqrt{\pi}}=\int_{0}^\pi \frac{d\phi}{\sqrt{\sin^4(\phi)+\cos^4(\phi)}}$$Using $u=\cos(\phi)$ I arrive at $$\frac{\Gamma^2(\frac{1}{4})}{2\sqrt{\pi}}=\int_{-1}^1 \frac{du}{\sqrt{(2u^4-2u^2+1)(1-u^2)}}$$
$$\frac{\Gamma^2(\frac{1}{4})}{4\sqrt{\pi}}=\int_{0}^1 \frac{du}{\sqrt{-2u^6+5u^4-3u^2+1}}$$
This looks to be similar to an elliptic integral but I am finding trouble reducing it to a form that I can evaluate.
EDIT: If I can evaluate the integral in terms of the Complete Elliptic Integral of the First Kind, I can use its relation with Jacobi's Third Theta Function to evaluate it in terms of $\eta$. Such that$$\frac{\Gamma^2(\frac{1}{4})}{4\sqrt{\pi}}=cK(k')=\frac{\pi}{2}\theta_3^2(q)$$
So that we arrive at the familiar form on the LHS $$\frac{\Gamma(\frac{1}{4})\pi^{-3/4}}{2}=\frac{\theta_3(q)}{\sqrt{2c}}$$
Best Answer
The key is the link between the Dedekind eta function and elliptic integrals.
Let $\tau$ be purely imaginary and in upper half of complex plane and let $$q=\exp(2\pi i\tau) \in(0,1)$$ be the corresponding nome. Consider the elliptic modulus $k\in(0,1)$ corresponding to nome $q$ given in terms of $q$ via Jacobi theta functions $$k=\frac{\vartheta_{2}^{2}(q)}{\vartheta _{3}^{2}(q)},\,\vartheta_{2}(q)=\sum_{n\in\mathbb {Z}} q^{(n+(1/2))^{2}},\,\vartheta _{3}(q)=\sum_{n\in\mathbb {Z}} q^{n^2}\tag{1}$$ Let $k'=\sqrt {1-k^2}$ and we further define elliptic integrals $$K=K(k) =\int_{0}^{\pi/2}\frac{dx} {\sqrt{1-k^2\sin^2 x}}, \, K'=K(k') \tag{2}$$ The circle of these definitions is finally completed by the formula $$\frac{K'} {K} =-2i\tau\tag{3}$$ Let $\tau'$ be another purely imaginary number in upper half of the complex plane such that $$\frac{\tau'} {\tau} =r\in\mathbb {Q} ^{+} \tag{4}$$ Let the corresponding nome be $q'=\exp(2\pi i\tau') $ and the elliptic moduli be $l, l'=\sqrt{1-l^2}$ and the elliptic integrals based on these moduli be denoted by $L, L'$. Then from the relation $\tau'=r\tau$ we get via $(3)$ the modular equation $$\frac{L'} {L} =r\frac{K'} {K}, r\in\mathbb {Q} ^{+} \tag{5}$$ Under these circumstances Jacobi proved using the transformation of elliptic integrals that the relation between moduli $k, l$ is algebraic and the ratio $K/L$ is an algebraic function of $k, l $.
The Dedekind's eta function is related to elliptic integrals via the relation $$\eta(\tau) =q^{1/24}\prod_{n=1}^{\infty} (1-q^n)=2^{-1/6}\sqrt{\frac{2K}{\pi}}k^{1/12}k'^{1/3}\tag{6}$$ Now let $\tau=i/2$ so that $q=e^{-\pi} $ and then from $(3)$ we have $K=K'$ so that $k=k'=1/\sqrt{2}$ and it is well known that for this value of $k$ we have $$K(k) =\frac{\Gamma^{2}(1/4)} {4\sqrt{\pi}} \tag{7}$$ From $(6)$ it now follows that $\eta(\tau) =\eta(i/2)$ is an algebraic multiple of $\Gamma (1/4)\pi^{-3/4}$.
Let $\tau'=ri, r\in \mathbb {Q} ^{+} $ so that $\tau'/\tau=2r$ is a positive rational number. As noted above if $l, L$ correspond to $\tau'$ then the relation between $l$ and $k=1 /\sqrt{2}$ is algebraic so that $l$ is an algebraic number and the ratio $K/L$ is an algebraic function of $k, l $ and thus $K/L$ is also an algebraic number. Thus from equation $(6)$ it follows that $\eta(ri) $ is an algebraic multiple of $\Gamma (1/4)\pi^{-3/4}$.
More generally it can be proved that if $r$ is a positive rational number then the value of $\eta(i\sqrt{r}) $ can be expressed in terms of values of Gamma function at rational points and $\pi$ and certain algebraic numbers.
Also let me complete the link between $\Gamma (1/4)$ and elliptic integrals starting with your approach. We have $$\frac{\Gamma ^2(1/4)}{2\sqrt{\pi}}=\int_{0}^{\pi}\frac{dx}{\sqrt{\sin^4 x+\cos^4 x}}=\int_{0}^{\pi}\frac{dx}{\sqrt{1-2\sin^2 x\cos^2 x}}$$ and the integral can further be written as $$\int_{0}^{\pi}\frac{dx}{\sqrt{1-(1/2)\sin^2 2x}}$$ Putting $2x =t$ we can see that it reduces to $$\frac{1}{2}\int_{0}^{2\pi}\frac{dt}{\sqrt{1-(1/2)\sin^2 t}}=2\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-(1/2)\sin^2 x}}=2K(1/\sqrt{2})$$ and we are done.