Euler’s Formula From Taylor Series’

The Taylor Series

Taylor polynomials are used to approximate function behavior around some domain value, c. When a real or complex function is infinitely differentiable and equal to an infinite Taylor series, it is known as an analytic function.

$f(x)=f(c)+f'(c)\frac{(x-c)^1}{1!}+f''(c)\frac{(x-c)^2}{2!}+...=\sum\limits_{n=0}^{\infty}f^{(n)}(c)\frac{(x-c)^n}{n!}$

When the function is evaluated at c=0, it is called a Maclaurin series.

$f(x)=f(0)+f'(0)\frac{x^1}{1!}+f''(0)\frac{x^2}{2!}+f'''(0)\frac{x^3}{3!}+...=\sum\limits_{n=0}^{\infty}f^{(n)}(0)\frac{x^n}{n!}$

Maclaurin Series of Cosine

To find the Maclaurin series representations of a function, we’ll start taking some derivatives and evaluating them at x=0

The cosine function and 6 of its derivatives evaluated at f(0), which repeats every 4 derivatives

f'(0) — The cosine function and 6 of its derivatives evaluated at f(0), which repeats every 4 derivatives

$f(0)$	$f'(0)$	$f''(0)$	$f'''(0)$	$f^{(4)}(0)$	$f^{(5)}(0)$	$f^{(6)}(0)$
$cos(0)=1$	$-sin(0)=0$	$-cos(0)=-1$	$sin(0)=0$	$cos(0)=1$	$-sin(0)=0$	$-cos(0)=-1$

$cos(x)=1-0\cdot\frac{x^1}{1!}-1\cdot\frac{x^2}{2!}+0\cdot\frac{x^3}{3!}+1\cdot\frac{x^4}{4!}+0\cdot\frac{x^5}{5!}-1\cdot\frac{x^6}{6!}+...\\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...=\sum\limits_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}$

The general cosine function and 6 of its derivatives evaluated at f(0), with coefficients

f'(0) — The general cosine function and 6 of its derivatives evaluated at f(0), with coefficients

$f(0)$	$f'(0)$	$f''(0)$	$f'''(0)$	$f^{(4)}(0)$	$f^{(5)}(0)$	$f^{(6)}(0)$
$a_1cos(0b_1)=a_1$	$-a_1b_1sin(0b_1)=0$	$-a_1b_1^2cos(0b_1)=-a_1b_1^2$	$a_1b_1^3sin(0b_1)=0$	$a_1b_1^4cos(0b_1)=a_1b_1^4$	$-a_1b_1^5sin(0b_1)=0$	$-a_1b_1^6cos(0b_1)=-a_1b_1^6$

$a_1cos(b_1x)=a_1-a_1b_1^2\frac{x^2}{2!}+a_1b_1^4\frac{x^4}{4!}-a_1b_1^6\frac{x^6}{6!}+...\\a_1cos(b_1x)=\sum\limits_{n=0}^\infty(-1)^na_1\frac{(b_1x)^{2n}}{(2n)!}$

Maclaurin Series of Sine

Naturally, we look to do the same with the sine function

The sine function and 6 of its derivatives evaluated at f(0), which repeats every 4 derivatives

f'(0) — The sine function and 6 of its derivatives evaluated at f(0), which repeats every 4 derivatives

$f(0)$	$f'(0)$	$f''(0)$	$f'''(0)$	$f^{(4)}(0)$	$f^{(5)}(0)$	$f^{(6)}(0)$
$sin(0)=0$	$cos(0)=1$	$-sin(0)=0$	$-cos(0)=-1$	$sin(0)=0$	$cos(0)=1$	$-sin(0)=0$

$sin(x)=0+1\cdot\frac{x^1}{1!}+0\cdot\frac{x^2}{2!}-1\cdot\frac{x^3}{3!}+0\cdot\frac{x^4}{4!}+1\cdot\frac{x^5}{5!}+0\cdot\frac{x^6}{6!}+...\\sin(x)=\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}+...=\sum\limits_{n=0}^\infty(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}$

The sine function and 6 of its derivatives evaluated at f(0), with coefficients

f'(0) — The sine function and 6 of its derivatives evaluated at f(0), with coefficients

$f(0)$	$f'(0)$	$f''(0)$	$f'''(0)$	$f^{(4)}(0)$	$f^{(5)}(0)$	$f^{(6)}(0)$
$a_2sin(0b_2)=0$	$a_2b_2cos(0b_2)=a_2b_2$	$-a_2b_2^2sin(0b_2)=0$	$-a_2b_2^3cos(0b_2)=-a_2b_2^3$	$a_2b_2^4sin(0b_2)=0$	$a_2b_2^5cos(0b_2)=a_2b_2^5$	$-a_2b_2^6sin(0b_2)=0$

$a_2sin(b_2x)=a_2b_2\frac{x^1}{1!}-a_2b_2^3\frac{x^3}{3!}+a_2b_2^5\frac{x^5}{5!}+...=\sum\limits_{n=0}^\infty(-1)^{n}a_2\frac{(b_2x)^{2n+1}}{(2n+1)!}$

Maclaurin Series of the Exponential

The exponential function and its derivatives, evaluated at f(0)

f'(0) — The exponential function and its derivatives, evaluated at f(0)

$f(0)$	$f'(0)$	$f''(0)$	$f'''(0)$	$f^{(4)}(0)$	$f^{(5)}(0)$	$f^{(6)}(0)$
$e^0=1$	$e^0=1$	$e^0=1$	$e^0=1$	$e^0=1$	$e^0=1$	$e^0=1$

$e^x=1+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+...=\sum\limits_{n=0}^\infty\frac{x^{n}}{n!}$

f(0) — The exponential function and its derivatives, with coefficients

f'(0) — The exponential function and its derivatives, with coefficients

$f(0)$	$f'(0)$	$f''(0)$	$f'''(0)$	$f^{(4)}(0)$	$f^{(5)}(0)$	$f^{(6)}(0)$
$a_3e^{b_30}=a_3$	$a_3b_3e^{b_30}=a_3b_3$	$a_3b_3^2e^{b_30}=a_3b_3^2$	$a_3b_3^3e^{b_30}=a_3b_3^3$	$a_3b_3^4e^{b_30}=a_3b_3^4$	$a_3b_3^5e^{b_30}=a_3b_3^5$	$a_3b_3^6e^{b_30}=a_3b_3^6$

$a_3e^{b_3x}=a_3+a_3b_3\frac{x^1}{1!}+a_3b_3^2\frac{x^2}{2!}+a_3b_3^3\frac{x^3}{3!}+a_3b_3^4\frac{x^4}{4!}=\sum\limits_{n=0}^\infty a_3\frac{(b_3x)^{n}}{n!}$

Putting Them Together

The exponential function and cosine/sine series’ contain all of the same terms, though with differing alternating signs.

$a_1cos(b_1x)=a_1-a_1b_1^2\frac{x^2}{2!}+a_1b_1^4\frac{x^4}{4!}-a_1b_1^6\frac{x^6}{6!}+...$

$a_2sin(b_2x)=a_2b_2\frac{x^1}{1!}-a_2b_2^3\frac{x^3}{3!}+a_2b_2^5\frac{x^5}{5!}+...$

$a_3e^{b_3x}=a_3+a_3b_3\frac{x^1}{1!}+a_3b_3^2\frac{x^2}{2!}+a_3b_3^3\frac{x^3}{3!}+a_3b_3^4\frac{x^4}{4!}+...$

We can begin to match variables up with each other, firstly by seeing that . Then, for simplicity, letting .

$cos(b_1x)=1-b_1^2\frac{x^2}{2!}+b_1^4\frac{x^4}{4!}-b_1^6\frac{x^6}{6!}+...$

$a_2sin(b_2x)=a_2b_2\frac{x^1}{1!}-a_2b_2^3\frac{x^3}{3!}+a_2b_2^5\frac{x^5}{5!}+...$

$e^{b_3x}=1+b_3\frac{x^1}{1!}+b_3^2\frac{x^2}{2!}+b_3^3\frac{x^3}{3!}+b_3^4\frac{x^4}{4!}+...$

Next, we can look at the terms, . Then, letting yields .

$cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...$

$a_2sin(b_2x)=a_2b_2\frac{x^1}{1!}-a_2b_2^3\frac{x^3}{3!}+a_2b_2^5\frac{x^5}{5!}+...$

$e^{ix}=1\pm i\frac{x^1}{1!}-\frac{x^2}{2!}\mp i\frac{x^3}{3!}+\frac{x^4}{4!}+...$

Since there are now multiple options for , we can compare and . If . Similarly, if . Moving forward with the reference angle, let and with the principal solution, .