Complex Power/Examples/(1 + i \tan (4m+1 over 4n) pi)^n

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.

$\blacksquare$