Hello StackExchange community,
this is probably a dumb question, but suppose you have a function that have the following power series
$f= \sum_{i = 0}^{\infty} f_i r^i$.
where $f:\mathbb{R} \rightarrow \mathbb{R}$, and $f_i$ are real constant coefficients. Is it possible to obtain asymptotic expansion of the following form
$f = \sum_{i = 0}^{\infty} \hat{f}_i r^{c\,i}$
where $c$ is just a constant?
Backstory: I'm solving Dirichlet boundary condition for Laplace equation in 2 dimensions. Asymptotic expansion of the solution in polar coordinates is known in this case
$u = \sum_{j = 1}^{\infty} r^{\lambda_j} \left(a_j\, \cos({\lambda_j \theta}) + b_j \sin (\lambda_j \theta)\right), \quad \lambda_j = j \frac{\pi}{\Theta}$
with $\Theta$ being angle between boundaries. Now I want to find coefficients $a_j$, $b_j$ by applying to boundary conditions, say $f\vert_{\theta_1}, g\vert_{\theta_2}$ in the following way
$r^{\lambda_j} \begin{bmatrix} \cos(\lambda_j\theta_1) & \sin(\lambda_j\theta_1) \\ \cos(\lambda_j\theta_2) & \sin(\lambda_j\theta_2) \\ \end{bmatrix} \begin{bmatrix} a_j \\ b_j \end{bmatrix} = r^{\lambda_j} \begin{bmatrix} f_j \\ g_j \end{bmatrix}$.
That is my motivation. If expansions for boundary conditions are not given in $\lambda_j$ I cannot do this. Or can I do something similar?