Euler's Secant Identity
Theorem
- $\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$
where:
- $z \in \C$ is a complex number
- $\sec z$ denotes the secant function
- $i$ denotes the imaginary unit: $i^2 = -1$
Proof
| \(\ds \sec z\) | \(=\) | \(\ds \frac 1 {\cos z}\) | Definition of Complex Secant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 / \frac {e^{i z} + e^{-i z} } 2\) | Euler's Sine Identity and Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 2 {e^{i z} + e^{-i z} }\) | multiplying top and bottom by $2$ |
$\blacksquare$
Also see
- Euler's Sine Identity
- Euler's Cosine Identity
- Euler's Tangent Identity
- Euler's Cotangent Identity
- Euler's Cosecant 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: $7.21$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$