A First Course in Complex Analysis · Problem 44

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-logarithm-equations

Problem

Find all solutions to the following equations: ((a))

\operatorname{Log}(z) = \frac{\pi i}{2}

Answer.

z = i

((b))

\operatorname{Log}(z) = \frac{3\pi i}{2}

Answer. there is no solution ((c))

\exp(z) = \pi i

Answer.

z = \ln\pi + i(\frac{\pi}{2} + 2\pi k),\ k \in \mathbb{Z}

((d))

\sin(z) = \cosh(4)

Answer.

z = \frac{\pi}{2} + 2\pi k \pm 4i,\ k \in \mathbb{Z}

((e))

\cos(z) = 0

Answer.

z = \frac{\pi}{2} + \pi k,\ k \in \mathbb{Z}

((f))

\sinh(z) = 0

Answer.

z = \pi k i,\ k \in \mathbb{Z}

((g))

\overline{\exp(iz)} = \exp(i\bar{z})

Answer.

z = \pi k,\ k \in \mathbb{Z}

((h))

z^{\frac{1}{2}} = 1+i

Answer.

z = 2i