Sum of Exponential of i k x

Theorem

$\ds \sum_{k \mathop = 0}^n \map \exp {i k x} = \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}$

where $x$ is a complex number that is not an integer multiple of $2 \pi$.

Proof

\(\ds \sum_{k \mathop = 0}^n \map \exp {i k x}\)

\(=\)

\(\ds \frac {\map \exp {i x \paren {n + 1} } - 1} {\map \exp {i x} - 1}\)

Sum of Geometric Sequence

\(\ds \)

\(=\)

\(\ds \frac {\map \exp {\frac {i x \paren {n + 1} } 2} \paren {\map \exp {\frac {i x \paren {n + 1} } 2} - \map \exp {\frac {-i x \paren {n + 1} } 2} } } {\map \exp {\frac {i x} 2} \paren {\map \exp {\frac {i x} 2} - \map \exp {-\frac {i x} 2} } }\)

factoring $\dfrac {\map \exp {\frac {i x \paren {n + 1} } 2} } {\map \exp {\frac {i x} 2} }$

\(\ds \)

\(=\)

\(\ds \map \exp {\frac {i x n} 2} \frac {2 i \map \sin {\frac {\paren {n + 1} x} 2} } {2 i \sin \frac x 2}\)

Exponential of Sum of Real Numbers: Corollary, Euler's Sine Identity

\(\ds \)

\(=\)

\(\ds \paren {i \sin \frac {n x} 2 + \cos \frac {n x} 2} \frac {\map \sin {\frac {\paren {n + 1} x} 2} } {\sin \frac x 2}\)

Euler's Formula

$\blacksquare$