Secant Exponential Formulation

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Theorem

Let $z$ be a complex number.

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

Then:

$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$


Proof

\(\displaystyle \sec z\) \(=\) \(\displaystyle \frac 1 {\cos z}\) Definition of Complex Secant Function
\(\displaystyle \) \(=\) \(\displaystyle 1 / \frac {e^{i z} + e^{-i z} } 2\) Sine Exponential Formulation and Cosine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac 2 {e^{i z} + e^{-i z} }\) multiplying top and bottom by $2$

$\blacksquare$


Also see


Sources