Let $ r > 0 $ and let $\vec{\gamma} = (\gamma_i)_{i=1}^2 $ such that $\gamma_{1} > 0 $, $\gamma_2 > 0 $ and such that $ \gamma_1 + \gamma_2 < 1$. We consider the following integral:
\begin{equation} J(r,\vec{\gamma}) := \int\limits_0^{2 \pi} \frac{\left( e^{r \cos(\phi)}-1, e^{r \sin(\phi)}-1 \right)}{1+\gamma_1\cdot (e^{r \cos(\phi)}-1) + \gamma_2 \cdot (e^{r \sin(\phi)}-1) } d\phi \end{equation}
This integral appears in the context of logarithmic utility portfolio optimization as in 1.
We have found a series expansion of this integral for small values of $r$. It reads:
\begin{equation} J(r,\vec{\gamma}) = \sum\limits_{m=1}^\infty r^{2 m} \cdot \sum\limits_{n=0}^{2 m} (-1)^n \binom{n}{p} \gamma_1^{n-p} \gamma_2^p \cdot \left(C^{(1,m)}_{n-p,p}, C^{(2,m)}_{n-p,p} \right) \quad (i) \end{equation}
where
\begin{eqnarray} C^{(1,m)}_{n-p,p} &:=& \sum\limits_{l_1=0}^{n-p+1} \sum\limits_{l_2=0}^p \binom{n-p+1}{l_1} \binom{p+0}{l_2} (-1)^{n+1-l_1-l2} \cdot \frac{(l_1^2+l_2^2)^m}{(m!)^2 \cdot 2^{2 m}} \\ C^{(2,m)}_{n-p,p} &:=& \sum\limits_{l_1=0}^{n-p+1} \sum\limits_{l_2=0}^p \binom{n-p+0}{l_1} \binom{p+1}{l_2} (-1)^{n+1-l_1-l2} \cdot \frac{(l_1^2+l_2^2)^m}{(m!)^2 \cdot 2^{2 m}} \end{eqnarray}
Now, there is an obvious question whether the series $(i)$ converge. I have tested it numerically . Below I show the number of terms that need to be taken into account in order to achieve an accuracy of nine decimal digits. In here I used $(\gamma_1,\gamma_2) = (0.1,0.2)$ (top-left), $(\gamma_1,\gamma_2) = (0.25,0.35)$ (top-right), $(\gamma_1,\gamma_2) = (0.333,0.4333)$ (bottom-left) and $(\gamma_1,\gamma_2) = (0.6,0.3)$ (bottom-right). We have:
As we can see the convergence rate slows down as the $\gamma_1+\gamma_2$ approaches unity from below and as $r$ increases yet even in the worst case scenario, i.e. $(\gamma_1,\gamma_2) = (0.6,0.3)$ and $r=1.4$ we do not need more than twenty terms in order to achieve the required precision.
Now, the question that I wanted to ask is how would you be computing this integral for big values of $r$ ? Is there a similar series expansion that one can come up with ?
1 Paragraph (4), page 38 in: T Goll, J Kallsen, Optimal portfolios for logarithmic utility, Stochastic Processes and their Applications 89 (2000), 31-48
This is going to be a partial answer to this question only.
First of all let us give the large $r$ limit of the integrand. We define: \begin{equation} f(r,\phi):= \frac{\left( e^{r \cos(\phi)}-1, e^{r \sin(\phi)}-1 \right)}{1+\gamma_1\cdot (e^{r \cos(\phi)}-1) + \gamma_2 \cdot (e^{r \sin(\phi)}-1) } d\phi \quad (i) \end{equation} and we have:
\begin{eqnarray} \lim_{r\rightarrow \infty} f(r,\phi) = \left( \begin{array}{r} \frac{1}{\gamma_1} \cdot 1_{-\pi/2 < \phi < \pi/4} - \frac{1}{1-\gamma_1-\gamma_2} \cdot 1_{\pi < \phi < 3/2 \pi} \\ \frac{1}{\gamma_2} \cdot 1_{+\pi/4 < \phi < \pi} - \frac{1}{1-\gamma_1-\gamma_2} \cdot 1_{\pi < \phi < 3/2 \pi} \end{array} \right) \end{eqnarray}
Below we can see a plot of the integrand $(i)$ for fixed $r = 100$ and $\gamma_1,\gamma_2 = 0.6,0.3 $. We conclude that the large $r$ limit is achieved very fast.
Therefore we conclude that:
\begin{equation} \lim_{r\rightarrow \infty } J(r,\vec{\gamma}) = \left( \begin{array}{r} \frac{1}{\gamma_1} \cdot \frac{3}{4} \pi - \frac{1}{1-\gamma_1-\gamma_2} \cdot \frac{\pi}{2} \\ \frac{1}{\gamma_2} \cdot \frac{3}{4} \pi - \frac{1}{1-\gamma_1-\gamma_2} \cdot \frac{\pi}{2} \end{array} \right) \end{equation}
Now let us look at the finite-$r$ corrections.
We define: \begin{eqnarray} \lambda^{\phi_-,\phi_+}_{r} &:=& \int\limits_{\phi_-}^{\phi_+} e^{-r \cdot \sin(\phi)} d\phi \\ &=& \frac{1}{r} \sum\limits_{m=0} \binom{m-1/2}{m} \frac{1}{r^{2 m}} \cdot \gamma(2m+1,x_1,x_+ ) \end{eqnarray} where $x_- := r \sin(\phi_-) \wedge r \sin(\phi_+) $ and $x_- := r \sin(\phi_-) \vee r \sin(\phi_+) $ .
We split our integrals accordingly.
\begin{eqnarray} J_1(r,\vec{\gamma}) &=& \underbrace{\int\limits_{-\pi/2}^{\pi/4} f_1(r,\phi) d\phi}_{{\mathcal J}_{1,a}} + \underbrace{\int\limits_{+\pi/4(1+\epsilon)}^{\pi(1-\epsilon} f_1(r,\phi) d\phi}_{{\mathcal J}_{1,b}} + \underbrace{\int\limits_{\pi(1+\epsilon)}^{3\pi/2(1-\epsilon)} f_1(r,\phi) d\phi}_{{\mathcal J}_{1,c}} \\ J_2(r,\vec{\gamma}) &=& \underbrace{\int\limits_{-\pi/2}^{\pi/4} f_2(r,\phi) d\phi}_{{\mathcal J}_{2,a}} + \underbrace{\int\limits_{+\pi/4}^{\pi} f_2(r,\phi) d\phi}_{{\mathcal J}_{2,b}} + \underbrace{\int\limits_{\pi}^{3\pi/2} f_2(r,\phi) d\phi}_{{\mathcal J}_{2,c}} \end{eqnarray}
\begin{eqnarray} {\mathcal J}_{1,a} &=& \frac{1}{\gamma_1} \cdot \int\limits_{-\pi/2}^{\pi/4} \left( \frac{1-e^{-r \cos(\phi)}}{1 + c_1 e^{-r \cos(\phi)} + c_2 e^{-r \sqrt{2} \cos(\phi+\pi/4)}} \right) d\phi \\ &=& \frac{1}{\gamma_1} \cdot \sum\limits_{n=0}^\infty (-1)^n \sum\limits_{p=0}^n \binom{n}{p} c_1^p \cdot c_2^{n-p} \left( %\int\limits_{\theta_{n,p} - \pi/4}^{\theta_{n,p} +\pi/2} %e^{-r \sqrt{n^2+(n-p)^2} \cdot \sin(\phi) } d\phi \lambda^{\theta_{n,p} - \pi/4,\theta_{n,p} +\pi/2}_{r \sqrt{n^2+(n-p)^2}} - %\int\limits_{\bar{\theta}_{n,p} - \pi/4}^{\bar{\theta}_{n,p} +\pi/2} %e^{-r \sqrt{(n+1)^2+(n-p)^2} \cdot \sin(\phi) } d\phi \lambda^{\bar{\theta}_{n,p} - \pi/4,\bar{\theta}_{n,p} +\pi/2}_{r \sqrt{(n+1)^2+(n-p)^2}} \right) \end{eqnarray}
\begin{eqnarray} {\mathcal J}_{1,b} &=& \frac{1}{\gamma_2} \cdot \int\limits_{\pi/4(1+\epsilon)}^{\pi(1-\epsilon)} \left( \frac{ e^{r \sqrt{2} \cos(\phi+\pi/4)}- e^{-r \sin(\phi)} }{ 1 + c_1 e^{-r \sin(\phi)} + c_2 e^{r \sqrt{2} \cos(\phi+\pi/4)} } \right) d\phi \\ &=& \frac{1}{\gamma_2} \cdot \sum\limits_{n=0}^\infty (-1)^n \sum\limits_{p=0}^n \binom{n}{p} c_1^p \cdot c_2^{n-p} \left( \lambda^{\pi/4(1+\epsilon) -\theta_{n,p},\pi(1-\epsilon) -\theta_{n,p}}_{r \sqrt{(n+1)^2+(n+1-p)^2}} - \lambda^{\pi/4(1+\epsilon)-\bar{\theta}_{n,p},\pi(1-\epsilon) - \bar{\theta}_{n,p}}_{r \sqrt{(n+1)^2+(n+0-p)^2}} \right) \end{eqnarray}
\begin{eqnarray} {\mathcal J}_{1,c} &=& \frac{1}{1-\gamma_1-\gamma_2} \cdot \int\limits_{\pi(1+\epsilon)}^{3/2\pi(1-\epsilon)} \left( \frac{ -1+ e^{r \cos(\phi)} }{ 1 + c_1 e^{r \cos(\phi)} + c_2 e^{r \sin(\phi)} } \right) d\phi \\ &=& \frac{1}{1-\gamma_1-\gamma_2} \cdot \sum\limits_{n=0}^\infty (-1)^n \sum\limits_{p=0}^n \binom{n}{p} c_1^p \cdot c_2^{n-p} \left( \lambda^{-\pi(1+\epsilon) -\theta_{n,p},-3/2\pi(1-\epsilon) -\theta_{n,p}}_{r \sqrt{(p+0)^2+(n-p)^2}} - \lambda^{-\pi(1+\epsilon)-\bar{\theta}_{n,p},-3/2\pi(1-\epsilon) - \bar{\theta}_{n,p}}_{r \sqrt{(p+1)^2+(n-p)^2}} \right) \end{eqnarray}