# The natural exponential function

This page is a sub-page of our page on Calculus of One Real Variable.

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The natural exponential function $\, e^x \,$ is the only function
that is unchanged by both differentiation and integration
.

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$\, \begin{matrix} D_{erivative} & F_{unction} & I_{ntegral} \\ & & & & & \\ \; 0 & \; 0 & \; 1 \\ & & & & & \\ \; 0 & \; 1 & \; x \\ & & & & & \\ \; 1 & \; x & \; \dfrac{1}{2} x^2 \\ & & & & & \\ \; x & \; \dfrac{1}{2} x^2 & \; \dfrac{1}{6} x^3 \\ & & & & & \\ \; \dfrac{1}{2} x^2 & \; \dfrac{1}{6} x^3 & \; \dfrac{1}{24} x^4 \\ & & & & & \\ \; \dfrac{1}{6} x^3 & \; \dfrac{1}{24} x^4 & \; \dfrac{1}{120} x^5 \\ & & & & & \\ \; \vdots & \; \vdots & \; \vdots \\ + \, \text{-----} & + \, \text{-----} & + \, \text{-----} \\ \;\; e^x & \;\; e^x & \;\; e^x \end{matrix} \,$

$\, e^x \, \equiv \, \displaystyle\sum_{n=0}^{\infty} \, \dfrac{x^n}{n \, !} \, \equiv \, 1 + x + \dfrac{1}{2} x^2 + \dfrac{1}{6} x^3 + \dfrac{1}{24} x^4 + \dfrac{1}{120} x^5 + \cdots \,$

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$\, \begin{matrix} D_{erivative} & F_{unction} & I_{ntegral} \\ & & & & & \\ i e^{i x} & e^{i x} & \dfrac{1}{i} e^{i x} \\ & & & & & \\ e^{i x} & \dfrac{1}{i} e^{i x} & \dfrac{1}{i^2} e^{i x} \\ & & & & & \\ \dfrac{1}{i} e^{i x} & \dfrac{1}{i^2} e^{i x} & \dfrac{1}{i^3} e^{i x} \\ & & & & & \\ \dfrac{1}{i^2} e^{i x} & \dfrac{1}{i^3} e^{i x} & \dfrac{1}{i^4} e^{i x} \\ & & & & & \\ \dfrac{1}{i^3} e^{i x} & \dfrac{1}{i^4} e^{i x} & \dfrac{1}{i^5} e^{i x} \end{matrix} \,$

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How (and why) to raise e to the power of a matrix (Steven Strogatz on YouTube)

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