Task
I am trying to solve Exercise 2.19 from [1]:
Suppose $(X_1, X_2)$ is a S$\alpha$S vector in $\mathbb{R}^2$ with uniform spectral measure $\Gamma$ such that $\Gamma(S_2) = 1$. Use polar coordinates to evaluate the joint characteristic function of $(X_1, X_2)$.
My attempt
The characteristic function of $(X_1, X_2)$ should be given by $$ \varphi(s_1, s_2) = \exp\bigg\{ - \int_{S_2} \vert s_1t_1 + s_2t_2 \vert^\alpha \,\,\Gamma(dt) \bigg\}, \quad s_1, s_2 \in \mathbb{R} $$ where $\Gamma$ is the unique, finite, symmetric spectral measure $\Gamma$ on the unit circle $S_2$. Consequently, using polar coordinates to transform the exponent gives us $$ \int_{S_2} \vert s_1t_1 + s_2t_2 \vert^\alpha \,\,\Gamma(dt) = \frac{1}{2\pi}\int_{0}^{2\pi} \vert s_1\cos(\psi) + s_2\sin(\psi) \vert^\alpha \,\,d\psi. $$ However, now I am at a loss of how to evaluate this integral. From Wikipedia's entry on isotropic multivariate stable vectors, I know that the result should go along the lines of $$ \varphi(s_1, s_2) = \exp \{ -\gamma_0^\alpha \vert s_1^2 + s_2^2 \vert^{\alpha / 2} \}, $$ where $\gamma_0$ is some positive constant. Using WolframAlpha shows me that my substitution gives the correct result at least for $\alpha = 2$ (possibly modulo normalization). So, I think the substitution is not wrong but nevertheless I do not know how to compute further steps.
Literature
[1] Samorodnitsky, G., & Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance: Stochastic Modeling (1st ed.). Routledge. https://doi.org/10.1201/9780203738818