(a) State the integral expression for the gamma function $\Gamma(z)$, for $\operatorname{Re}(z)>0$, and express the integral

$$\int_{0}^{\infty} t^{\gamma-1} e^{i t} d t, \quad 0<\gamma<1$$

in terms of $\Gamma(\gamma)$. Explain why the constraints on $\gamma$ are necessary.

(b) Show that

$$\int_{0}^{\infty} \frac{e^{-k t^{2}}}{\left(t^{2}+t\right)^{\frac{1}{4}}} d t \sim \sum_{m=0}^{\infty} \frac{a_{m}}{k^{\alpha+\beta m}}, \quad k \rightarrow \infty$$

for some constants $a_{m}, \alpha$ and $\beta$. Determine the constants $\alpha$ and $\beta$, and express $a_{m}$ in terms of the gamma function.

State without proof the basic result needed for the rigorous justification of the above asymptotic formula.

[You may use the identity:

$$\left.(1+z)^{\alpha}=\sum_{m=0}^{\infty} c_{m} z^{m}, \quad c_{m}=\frac{\Gamma(\alpha+1)}{m ! \Gamma(\alpha+1-m)}, \quad|z|<1 .\right]$$