# Primitive of Exponential of a x/Complex

## Theorem

$\ds \int e^{a x} \rd x = \frac {e^{a x} } a + C$

## Proof for Complex Numbers

Let $z \in \C$ be a complex variable.

 $\ds \map {D_x} {\frac {e^{a z} } a}$ $=$ $\ds \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!} }$ Definition of Complex Exponential Function $\ds$ $=$ $\ds \map {D_x} {\frac 1 a \sum_{n \mathop = 0}^\infty \frac {a^n z^n} {n!} }$ Exponent Combination Laws $\ds$ $=$ $\ds \map {D_x} {\sum_{n \mathop = 0}^\infty \frac {a^{n - 1} z^n} {n!} }$ Summation is Linear: Scaling of Summations $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty n \frac {a^{n - 1} z^{n - 1} } {n!}$ Derivative of Complex Power Series $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty n \frac {\paren {a z}^{n - 1} } {n!}$ Exponent Combination Laws $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {\paren {a z}^{n - 1} } {\paren {n - 1}!}$ $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {\paren {a z}^n} {n!}$ Translation of Index Variable of Summation $\ds$ $=$ $\ds e^{a z}$ Definition of Complex Exponential Function

The result follows by the definition of the primitive.

$\blacksquare$