I'm trying to prove a result from Leon Cohen's book "Time-frequency Analysis", Chapter 18. Namely, I want to verify the solution of the eigenvalue problem $$\mathcal{C}s(t) = c s(t)$$ for the scale operator $\mathcal{C}$, defined by $$\mathcal{C}s(t) = \frac{1}{2j}\left(t \frac{\mathrm{d}}{\mathrm{d}t} + \frac{\mathrm{d}}{\mathrm{d}t}t \right)s(t).$$
After some manipulation, this amounts to solving for a function $\gamma(t)$ the following differential equation: $$\gamma'(t) = \left(j c - \frac{1}{2} \right) \frac{1}{t} \gamma(t),$$ where $c$ is a real constant, $t$ is real and $j = \sqrt{-1}$.
The book gives the solution (we don't care about initial values) $$\gamma(t) = \frac{1}{\sqrt{2\pi}} \frac{e^{j c \ln(t)}}{\sqrt{t}},$$ valid for $t > 0$. I want to verify this solution. Here's what I came up with.
$$\begin{align} \gamma'(t) &= \left(j c - \frac{1}{2} \right) \frac{1}{t} \gamma(t) \\ \frac{1}{\gamma}\frac{\mathrm{d}\gamma}{\mathrm{d}t} &= \left(j c - \frac{1}{2} \right) \frac{1}{t} \\ \int_{t_0}^t \frac{1}{\gamma}\frac{\mathrm{d}\gamma}{\mathrm{d}\tau} \mathrm{d} \tau &= \int_{t_0}^t \left(j c - \frac{1}{2} \right) \frac{1}{\tau} \mathrm{d} \tau \\ \int_{\gamma(t_0)}^{\gamma(t)} \frac{1}{\gamma} \mathrm{d} \gamma &= \left(j c - \frac{1}{2} \right) \int_{t_0}^t \frac{1}{\tau} \mathrm{d} \tau \\ \ln\left(\frac{|\gamma(t)|}{|\gamma(t_0)|}\right) &= \left(j c - \frac{1}{2} \right) (\ln(t) - \ln(t_0)),\quad \text{for}~t_0, t > 0 \\ \exp\left(\ln\left(\frac{|\gamma(t)|}{|\gamma(t_0)|}\right) \right) &= \exp\left( \left(j c - \frac{1}{2} \right) (\ln(t - \ln(t_0) \right) \\ \frac{|\gamma(t)|}{|\gamma(t_0)|} &= e^{\left(jc - \frac{1}{2} \right) [\ln(t) - \ln(t_0)]} \\ \gamma(t) &= \pm \gamma(t_0) e^{(jc - \frac{1}{2})\ln(t_0)} \frac{e^{jc \ln(t)}}{\sqrt{t}} \\ \gamma(t) &= K(t_0, \gamma(t_0)) \frac{e^{jc \ln(t)}}{\sqrt{t}}, \end{align}$$ where $K$ is a complex constant we can set to $\frac{1}{\sqrt{2\pi}}$. My question is about the step where I solve the integrals. For the right integral it's all good, $t$ is a real variable and I understand the definite integral converges to $\ln(|t|) - \ln(|t_0|)$ as long as both $t$ and $t_0$ have the same sign (to avoid integrating around $0$). My problem is with the left integral. Since $\gamma$ is complex, I'm taking a line integral over the complex plane, which as I understand can depend on the path, so I shouldn't be able to naively evaluate it to $\ln{|\gamma(t)|} - \ln{|\gamma(t_0)|}$ just using the endpoints. Am I correct in this assumption? If so, how can I fix this solution to be mathematically sound?
