Obtain solutions of the second-order ordinary differential equation

$$z w^{\prime \prime}-w=0$$

in the form

$$w(z)=\int_{\gamma} f(t) e^{-z t} d t$$

where the function $f$ and the choice of contour $\gamma$ should be determined from the differential equation.

Show that a non-trivial solution can be obtained by choosing $\gamma$ to be a suitable closed contour, and find the resulting solution in this case, expressing your answer in the form of a power series.

Describe a contour $\gamma$ that would provide a second linearly independent solution for the case $\operatorname{Re}(z)>0$.