Secant Exponential Formulation

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

\(\ds \sec z\)

\(=\)

\(\ds \frac 1 {\cos z}\)

Definition of Complex Secant Function

\(\ds \)

\(=\)

\(\ds 1 / \frac {e^{i z} + e^{-i z} } 2\)

Sine Exponential Formulation and Cosine Exponential Formulation

\(\ds \)

\(=\)

\(\ds \frac 2 {e^{i z} + e^{-i z} }\)

multiplying top and bottom by $2$

$\blacksquare$

Also see

• Sine Exponential Formulation
• Cosine Exponential Formulation
• Tangent Exponential Formulation
• Cotangent Exponential Formulation
• Cosecant Exponential Formulation

• Arcsecant Logarithmic Formulation

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.21$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$