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1

$e$ is just as important as $\pi$ in mathematics having uses in pretty much every field. For example, $$e=\lim\limits_{x\to \infty}\left(1+\dfrac{1}{x}\right)^x$$ One of the most beautiful examples of its importance would be relating trigonometric functions to hyperbolic functions using the identity: $$e^{ix} = \cos(x)+i\sin(x)$$ For example: ...

2

$e$ is fundamental in mathematics. Aside from the awesome properties of $e$, such as $e^{i \pi}+1=0$ and the fact that $$\frac{d}{dx} e^x=e^x,$$ it is also found in equations that directly relate to everyday phenomena. For instance, the normal distribution is represented by the probability density function $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}. ... 0 Eular's identity pops up everywhere, in calculus, differential equations and even probabilities. For example in elementary probability theory, it shows up in the Poisson distribution general formula which is used to calculate the probability of an event occurring given that we know the rate at which it happens$$P(X=k)=\frac{\lambda^k}{k!} e^{-\lambda}$$... 1 By expanding the (R-r)^2 piece, you see that you basically are looking to evaluate$$\int_s^R dr \frac{r^k}{\sqrt{r^2-s^2}}$$for k=1,2,3. For k=1, the integral is simple:$$\int_s^R dr \frac{r}{\sqrt{r^2-s^2}} = \frac12 \int_{s^2}^{R^2} \frac{du}{\sqrt{u-s^2}} = \sqrt{R^2-s^2}$$For k=2, integration by parts is useful:$$\begin{align}\int_s^R ...

0

From $e=\lim\limits_{n\to\infty}\left(1+\frac1n\right)^n$ easily follows $e^x=\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n$ for $x>0$. If you prove that even $e=\lim\limits_{n\to-\infty}\left(1+\frac1n\right)^n$, then $e^x=\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n$ also for $x<0$. Expand the expression inside the limit: \left(1+\frac ...

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