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.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Exercise $10$