Complex sine function ( $sin z$ )

The complex sine is the entire function

sin z = \frac{e ^{i z} - e ^{- i z}}{2 i} .

For real

x, y

,

sin (x + i y) = sin x cosh y + i cos x sinh y

.

Table of contents
- Periodicity of the complex sine Complex sine function

Periodicity of the complex sine ( $sin (z + 2 π) = sin z$ )

The kernel

2 πi Z

of the complex exponential function gives

sin (z + 2 π) = \frac{e ^{i z} e ^{2 πi} - e ^{- i z} e ^{- 2 πi}}{2 i} = sin z .

The same period appears as the translation generated by the monodromy of the complex inverse sine.

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Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 1 / 1C / ii / Solution

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