Show that the equation

$$z w^{\prime \prime}+w^{\prime}+(\lambda-z) w=0$$

has solutions of the form

$$w(z)=\int_{\gamma}(t-1)^{(\lambda-1) / 2}(t+1)^{-(\lambda+1) / 2} e^{z t} d t$$

Give examples of possible choices of $\gamma$ to provide two independent solutions, assuming $\operatorname{Re}(z)>0$. Distinguish between the cases $\operatorname{Re} \lambda>-1$ and $\operatorname{Re} \lambda<1$. Comment on the case $-1<\operatorname{Re} \lambda<1$ and on the case that $\lambda$ is an odd integer.

Show that, if $\operatorname{Re} \lambda<1$, there is a solution $w_{1}(z)$ that is bounded as $z \rightarrow+\infty$, and that, in this limit,

$$w_{1}(z) \sim A e^{-z} z^{(\lambda-1) / 2}\left(1-\frac{(1-\lambda)^{2}}{8 z}\right),$$

where $A$ is a constant.