 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 2

Write $z=re^{i\theta}$ with $r>0$ and $0\leq\theta<2\pi$, and write $w=u+iv$. Since
$$
e^w=e^ue^{iv}=re^{i\theta},
$$
we have
$$
u=\log r,
\qquad
v=\theta+2\pi n,
\qquad n\in\mathbb Z.
$$
Thus the inverse of the exponential is the infinitely valued <complex logarithm>
$$
\boxed{w(z)=\log r+i(\theta+2\pi n),\qquad n\in\mathbb Z.}
$$
For the angle convention in the question, its principal value is
$$
\boxed{\operatorname{Log}z=\log r+i\theta,\qquad0\leq\theta<2\pi.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / i / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / i / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)