Cosecant Exponential Formulation

Theorem

Let $z$ be a complex number.

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

Then:

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

Proof

\(\ds \csc z\)

\(=\)

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

Definition of Complex Cosecant Function

\(\ds \)

\(=\)

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

Sine Exponential Formulation

\(\ds \)

\(=\)

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

multiplying top and bottom by $2 i$

$\blacksquare$

Also see

• Sine Exponential Formulation
• Cosine Exponential Formulation
• Tangent Exponential Formulation
• Cotangent Exponential Formulation
• Secant Exponential Formulation

• Arccosecant Logarithmic Formulation

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.22$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$