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Exponential function
Definition
| $\mathrm{exp}: \mathbb C\to\mathbb C$ |
| $\mathrm{exp}(z):=\sum_{k=0}^\infty \frac{1}{k!} z^k $ |
Discussion
Definitions
Eulers number:
| $\mathrm{e}\equiv \mathrm{exp}(1)$ |
Theorems
| $\mathrm{e}^z = \mathrm{exp}(z) $ |
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Because per definition $\mathrm{e}^z:=\mathrm{exp}(z\cdot \mathrm{ln}(\mathrm{e}))$.
| $\mathrm{e}^z \neq 0 $ |
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For any polynomial $p$ and sufficiently often differentiable function $f$ we have
| $p\left(\frac{\mathrm d}{\mathrm d z}\right)\ \mathrm{e}^{f(z)} = p\left(\frac{\mathrm d}{\mathrm d z}\right)f(z)\cdot \mathrm{e}^{f(z)} $ |
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References
Wikipedia: Exponential function, Exponential map
Parents
Requirements
