Cotangent Exponential Formulation

Theorem

Let $z$ be a complex number.

Let $\cot z$ denote the cotangent function and $i$ denote the imaginary unit: $i^2 = -1$.

Then:

$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$

Proof 1

 $\displaystyle \cot z$ $=$ $\displaystyle \frac {\cos z} {\sin z}$ Definition of Complex Cotangent Function $\displaystyle$ $=$ $\displaystyle \frac {e^{i z} + e^{-i z} } 2 / \frac {e^{i z} - e^{-i z} } {2 i}$ Sine Exponential Formulation and Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle i \frac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$ multiplying numerator and denominator by $2 i$

$\blacksquare$

Proof 2

 $\displaystyle \cot z$ $=$ $\displaystyle \frac 1 {\tan z}$ Definition of Complex Cotangent Function $\displaystyle$ $=$ $\displaystyle 1 / \dfrac {e^{i z} - e^{-i z} } {i \left({e^{i z} + e^{-i z} }\right)}$ Tangent Exponential Formulation/Formulation 2 $\displaystyle$ $=$ $\displaystyle i \frac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$

$\blacksquare$

Also defined as

This result is sometimes also presented as:

$\cot z = \dfrac {i \paren {e^{i z} + e^{-i z} } } {e^{i z} - e^{-i z} }$