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5 problems match “complex exponentials”. Clear

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1. Integral of an exponential over a triangle ●○○○○ 10 ptsLet $\gamma$ be the triangle in $\mathbb C$ whose vertices are the three cube roots of $1+i$, oriented clockwise. Find the value of $$\int_\gamma e^{2z+i}\, dz.$$
2. Principal value of i to the i ●●○○○ 20 ptsFind the principal value of $i^{i}$, where $i=\sqrt{-1}$. Round to two decimal places.
3. Simplify a complex exponential ratio ●●○○○ 20 ptsLet $\theta$ be an angle such that $\tan \theta = 2$. Simplify $\frac{e^{2i \theta} - 1}{e^{2i \theta} + 1}$ and write the result in the form $a+bi$ with real numbers $a$ and $b$. Find $a$ and $b$.
4. Simplify expression with imaginary cosine and sine ●●●○○ 50 ptsSimplify $2\cos^2(i\log 2009)+i\sin(i\log 4036081)$, where $\log$ is the natural logarithm. Give your answer as a decimal rounded to eight decimal places.
5. Count complex solutions of an exponential equation ●●●●○ 100 ptsHow many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation $e^z = \frac{z - 1}{z + 1}$?