Complex Power/Examples/(1 + i \tan (4m+1 over 4n) pi)^n
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Example of Complex Power
For $m, n \in \Z$ such that $n \ne 0$:
- $\paren {1 + i \map \tan {\dfrac {4 m + 1} {4 n} \pi} }^n = \paren {-1}^m \paren {\sec \dfrac {4 m + 1} {4 n} \pi}^n \paren {\dfrac {1 + i} {\sqrt 2} }$
Proof
First setting $x = \dfrac {4 m + 1} {4 n} \pi$, we have:
\(\ds 1 + i \tan x\) | \(=\) | \(\ds 1 + i \frac {\sin x} {\cos x}\) | Definition of Real Tangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cos x + i \sin x} {\cos x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {1 + i \tan x}^n\) | \(=\) | \(\ds \frac {\cos n x + i \sin n x} {\cos^n x}\) | De Moivre's Theorem | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {1 + i \tan \paren {\dfrac {4 m + 1} {4 n} \pi} }^n\) | \(=\) | \(\ds \frac {\cos n \paren {\dfrac {4 m + 1} {4 n} \pi} + i \sin n \paren {\dfrac {4 m + 1} {4 n} \pi} } {\cos^n \paren {\dfrac {4 m + 1} {4 n} \pi} }\) | substituting back for $x$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \cos {\dfrac {4 m + 1} 4 \pi} + i \map \sin {\dfrac {4 m + 1} 4 \pi} } {\map {\cos^n} {\dfrac {4 m + 1} {4 n} \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \cos {m \pi + \dfrac \pi 4} + i \map \sin {m \pi + \dfrac \pi 4} } {\map {\cos^n} {\dfrac {4 m + 1} {4 n} \pi} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map \cos {m \pi + \dfrac \pi 4} + i \map \sin {m \pi + \dfrac \pi 4} } \paren {\map \sec {\dfrac {4 m + 1} {4 n} \pi} }^n\) | Definition of Secant Function |
Now we have:
\(\ds \map \cos {m \pi + \dfrac \pi 4} + i \map \sin {m \pi + \dfrac \pi 4}\) | \(=\) | \(\ds \paren {-1}^m \paren {\cos \dfrac \pi 4 + i \sin \dfrac \pi 4}\) | Sine and Cosine of Angle plus Integer Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^m \paren {\frac {\sqrt 2} 2 + i \frac {\sqrt 2} 2}\) | Cosine of $\dfrac \pi 4$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^m \paren {\dfrac {1 + i} {\sqrt 2} }\) | simplifying |
The result follows.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Exercise $10$