Euler's Cosecant Identity
Theorem
- $\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$
where:
- $z \in \C$ is a complex number
- $\csc z$ denotes the cosecant function
- $i$ denotes the imaginary unit: $i^2 = -1$
Proof
| \(\ds \csc z\) | \(=\) | \(\ds \frac 1 {\sin z}\) | Definition of Complex Cosecant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 / \frac {e^{i z} - e^{-i z} } {2 i}\) | Euler's Sine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 i} {e^{i z} - e^{-i z} }\) | multiplying top and bottom by $2 i$ |
$\blacksquare$
Also see
- Euler's Sine Identity
- Euler's Cosine Identity
- Euler's Tangent Identity
- Euler's Cotangent Identity
- Euler's Secant Identity
Source of Name
This entry was named for Leonhard Paul Euler.
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: $4$