Evaluate $\int_{\gamma} f(z) d z.$ $f(z)=z^{2}-i z, \gamma$ is the quarter circle about the origin from 2 to 2i.

Consider the initial value problem

$$y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0$$

where $\gamma$ is the damping coefficient (or resistance).

Determine how $t_1$ and $y_1$ vary as $\gamma$ decreases. What are the values of $t_1$ and $y_1$ when $\gamma=0?$

Prove that the function H defined in Eq. (10) is analytic inside $\Gamma$ and that its derivative is given by formula (11). $H ( z ) : = \int _ { \Gamma } \frac { g ( \zeta ) } { ( \zeta - z ) ^ { 2 } } d \zeta \quad ( z \text { not on } \Gamma )$, (10) $H ^ { \prime } ( z ) = 2 \int _ { \Gamma } \frac { g ( \zeta ) } { ( \zeta - z ) ^ { 3 } } d \zeta \quad ( z \text { not on } \Gamma )$, (11)

Evaluate $\int_{\gamma} f(z) d z.$ $f(z)=\overline{i z}, \gamma$ is the line segment from 0 to -4 + 3i.

Use the recursion relation, and if needed, equation to simplify: $\Gamma(2 / 3) / \Gamma(5 / 3)$

Evaluate $\int_{\gamma} f(z) d z.$ $f(z)=\operatorname{Re}(z), \gamma$ is the line segment from 1 to 2 + i.

Evaluate the following: $\Gamma(6)$.

Evaluate $\int_{\gamma} f(z) d z.$ $f(z)=1, \gamma(t)=t^{2}-i t \text { for } 1 \leq t \leq 3.$

Find a bound for $\left|\int_{\gamma} \cos \left(z^{2}\right) d z\right|$ if $\gamma$ is the circle of radius 4 about the origin.

When are gamma rays emitted?

Evaluate $\lim _{\varepsilon \rightarrow 0} \frac{F_{0}}{2 \varepsilon \gamma} \sin \varepsilon t \sin \gamma t.$

Find the solutions of the equation that are in the interval $[0,2 \pi)$.

$$2 \cos ^2 \gamma+\cos \gamma = 0$$

(a) What is $\gamma$ if v=0.250c? (b) If v=0.500c?

Use the known value $\Gamma\left(-\frac{1}{2}\right)=$ $-2 \sqrt{\pi}$ to evaluate the given quantity. $\Gamma\left(-\frac{5}{2}\right) 10.$\Gamma\left(-$\frac{9}{2}$\right)$