Euler's Cotangent Identity
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Theorem
- $\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$
where:
- $z \in \C$ is a complex number such that $\forall k \in \Z: z \ne k \pi$
- $\cot z$ denotes the cotangent function
- $i$ denotes the imaginary unit: $i^2 = -1$
Proof 1
We have, by hypothesis, that $z$ is a complex number such that:
- $\forall k \in \Z: z \ne k \pi$
Therefore:
- $\sin z \ne 0$
It follows from the definition of the complex cotangent function that:
- $\cot z$
is well-defined.
Hence:
\(\ds \cot z\) | \(=\) | \(\ds \frac {\cos z} {\sin z}\) | Definition of Complex Cotangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i z} + e^{-i z} } 2 / \frac {e^{i z} - e^{-i z} } {2 i}\) | Euler's Sine Identity and Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds i \frac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }\) | multiplying numerator and denominator by $2 i$ |
$\blacksquare$
Proof 2
We have, by hypothesis, that $z$ is a complex number such that:
- $\forall k \in \Z: z \ne k \pi$
Therefore:
- $\sin z \ne 0$
It follows from the definition of the complex cotangent function that:
- $\cot z$
is well-defined.
Hence:
\(\ds \cot z\) | \(=\) | \(\ds \frac 1 {\tan z}\) | Definition of Complex Cotangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 / \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }\) | Euler's Tangent Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds i \frac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }\) | Definition of Reciprocal |
$\blacksquare$
Also presented as
Euler's Cotangent Identity can also be presented as:
- $\cot z = \dfrac {i \paren {e^{i z} + e^{-i z} } } {e^{i z} - e^{-i z} }$
Also see
- Euler's Sine Identity
- Euler's Cosine Identity
- Euler's Tangent Identity
- Euler's Secant 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: $7.20$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$