Engineering Mathematics: Exponential Function

E.ID.1: Taylor Series

$e^x\ =\ 1\ +\ x\ +\ \frac{x^2}{2} +\ \frac{x^3}{6} +\ \frac{x^4}{24} +\ \frac{x^5}{120}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{x^n}{n!}$

E.ID.2: Euler's Formula

$e^{ix}\ =\ cosx\ +\ i sinx$

$cosx\ =\ \frac{1}{2}\left(e^{ix}\ +\ e^{-ix}\right)$

$i sinx\ =\ \frac{1}{2}\left(e^{ix}\ -\ e^{-ix}\right)$

E.ID.3: Bernouli's Formula

$e^x\ =\ \lim_{n\rightarrow\infty}\left(1\ +\ \frac{x}{n}\right)^n$

Proof

power expansion

$\left ( 1+\frac{x}{n} \right )^{n}=1+\binom{n}{1}\frac{x}{n}+\binom{n}{2}\frac{x^{2}}{n^{2}}+\binom{n}{3}\frac{x^{3}}{n^{3}}+\cdot \cdot \cdot +\binom{n}{n}\frac{x^{n}}{n^{n}}$

$\left ( 1+\frac{x}{n} \right )^{n}=1+\frac{n!}{1!(n-1)!}\frac{x}{n}+\frac{n!}{2!(n-2)!}\frac{x^{2}}{n^{2}}+\frac{n!}{3!(n-3)!}\frac{x^{3}}{n^{3}}+\cdot \cdot \cdot +\frac{n!}{n!0!}\frac{x^{n}}{n^{n}}$
after dividing

$\left ( 1+\frac{x}{n} \right )^{n}=1+\frac{n}{1!}\frac{x}{n}+\frac{n(n-1)}{2!}\frac{x^{2}}{n^{2}}+\frac{n(n-1)(n-2)}{3!}\frac{x^{3}}{n^{3}}+\cdot \cdot \cdot +\frac{n(n-1)(n-2)\cdot \cdot \cdot \cdot 1}{n!}\frac{x^{n}}{n^{n}}$

again , after dividing

$\left ( 1+\frac{x}{n} \right )^{n}=1+x+\frac{1(1-\frac{1}{n})}{2!}x^{2}+\frac{1(1-\frac{1}{n})(1-\frac{2}{n})}{3!}x^{3}+\cdot \cdot \cdot +\frac{1(1-\frac{1}{n})(1-\frac{2}{n})\cdot \cdot \cdot \cdot \frac{1}{n}}{n!}x^{n}$

$n \to \infty$

$\left ( 1+\frac{x}{n} \right )^{n}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdot \cdot \cdot +\frac{x^{n}}{n!}=e^{x}$
E.ID.4: Logarithms

$e^{ln(x)}\ =\ x$

$a^x\ =\ \left(e^{ln(a)}\right)^x\ =\ e^{x\,ln(a)}$

$y\ =\ a^x \Leftrightarrow\ x\ =\ log_a(y)\ \equiv\ \frac{ln(y)}{ln(a)}$

$\frac{da^x}{dx}\ =\ ln(a)\,e^{x\,ln(a)}\ =\ ln(a)\,a^x$