# Primitive of Exponential of a x/Complex

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## 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.

$\blacksquare$

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