Jtdylktuy

Here's a slick derivation. Let $f(x)= e^{-ix}(cos x + isin x)$. Taking the derivative we have that $f'(x) = -ie^{-ix}(cos x + i sin x) + e^{-ix}(-sin x + icos x) = 0$ so $f(x)$ is constant. But $f(0) = e^0(1 + 0) = 1$ so $f equiv 1$ so $e^{ix} = cos x + i sin x$. Plugging in $x = pi$ yields the result.

Remark: This proof is very beautiful, and uses the characteristic property of everything involved ($e^x$ is the unique nontrivial function that is its own derivative, $i$ squares to $-1$, etc), but as a first-timer the Taylor series proof told me much more.

The identity is a special case of Euler's formula from complex analysis, which states that

$e^{ix}$=$cos x+icdotsin x$

for any real number $x$. (Note that the variables of the trigonometric functions sine and cosine are taken to be in radians, and not in degrees.) In particular, with $x = pi$, or one half turn around the circle:

$e^{ipi}$=$cospi+icdotsinpi$

Since

$cospi=-1$

and

$sinpi=0$

it follows that

$e^{ipi}$ =$-1+0i$

which gives the identity

$e^{ipi}$$+1=0$

The physical explanation of Euler's identity is that it can be viewed as the group-theoretical definition of the number $pi$. The following discussion is at the physical level but can be made mathematically strict. The group is the group of rotations of a plane around 0. In fact, one can write:

$e^{ipi}$=$(e^{idelta})^{pi/delta}$

with $delta$ being some small angle.
The last equation can be seen as the action of consecutive small shifts along the circle caused by the application of infinitesimal rotations starting at 1 and going for the total length of the arc connecting points 1 and -1 in the complex plane. In fact, each small shift can be written as multiplication by

$1+idelta$

and the total number of shifts is π/δ. In order to get from 1 to -1 the total transformation would be

$(1+idelta)^{pi/delta}$

Now, taking the limit when $delta to0$, denoting $idelta = 1/n$ and using the definition of:

$e$=$limlimits_{n to infty}(1+frac{1}{n})^n$

we arrive at Euler's identity. The $pi$ itself is defined as the total angle which connects $1$ to $-1$ along the arch.

Summarizing, we can say that because the circle can be defined through the action of the group of shifts which preserve the distance between a point and another point, the relation between π and e arises.
This simple argument is the key to understanding other seemingly miraculous relations involving $π$ and $e$.

Source: Wikipedia

$$e^{ix}=sum_{n=0}^{infty}frac{(ix)^n}{n!}=sum_{k=0}^{infty}frac{(ix)^{2k}}{(2k)!}+sum_{k=0}^{infty}frac{(ix)^{2k+1}}{(2k+1)!}=$$
$$=sum_{k=0}^{infty}(-1)^kfrac{x^{2k}}{(2k)!}+isum_{k=0}^{infty}(-1)^kfrac{x^{2k+1}}{(2k+1)!}=$$
$$=cos x+isin x$$
since $$i^{2k}=(-1)^k,i^{2k+1}=i(-1)^k$$
for $x=pi$ we get
$$e^{ipi}=cos (pi)+isin(pi)=-1$$

1) The definition of powers $a^b$, where $b$ is an arbitrary rational number and $a>0$ is derived by the algebraic properties of exponents of positive integer exponents.

2) The definition of powers $a^b$, where $b$ is an arbitrary real number and $a>0$ is defined as the limit of powers with rational exponents: Choose a sequence of rational numbers $q_nto b$, and let $a^b=lim a^{q_n}$. (There are several things to prove here to show this makes sense, but it's all ok).

3) For the particular value $b=e$, one obtains the function $exp (x)=e^x$. This function is infinitely differentiable with $exp '(x)=exp (x)$. Thus it's Taylor expansion is particularly nice (and well-known).

4) The radius of convergence of the Taylor expansion of $exp$ is shown to be $infty $ and to converge everywhere to $exp(x)$. Thus, any complex number can be plugged into it, and so $exp(z)$ is defined for all complex numbers.

5) The trigonometric functions $sin$ and $cos$ can be defined as follows: The point $(cos x,sin x)$ is that point on the unit circle in the $X-Y$ plane that forms an angle of $x$ radians with the positive $X$-axis (measuring from the $X$-axis counter clockwise).

6) From geometric considerations it can be shown that these trigonomerric functions are infinitely differentiable, with their familiar derivatives, and with their familiar Taylor expansions.

7) The Taylor expansions have infinite radius of convergence, and converges everywhere to to their respective sructures. Thus $sin z$ and $cos z$ are defined for all complex numbers.

8) By the general theory of power series, the three power series mentioned above can be manipulated much like finite power series (order of summation changes without affecting the result etc.).

9) When plugging in $pi i$ into the power series expansion of $exp$, thus computing $e^{pi i}$, one can show (this is a good exercise) that: $$e^{pi i}=cos pi + icdot sin pi.$$

10) Thus $e^{pi i}=-1$.

Remark: More modern treatments of the trigonometric and exponential functions, in order to avoid unnecessary subtle geometric considerations and limiting mess, define these functions via their Taylor expansions.

It's important to note that this identity is dependent upon our idea of complex exponentiation, which is to say how we decide to extend the exponential function to the complex plane. By convention, this is done by declaring the exponential function to be analytic with the same Taylor series as it has in the reals. This is why derivations of this fact all use some argument which relies on Taylor expansion or at least differentiation acting the same way for complex numbers as they do for real numbers (and it seems like these two should be equivalent (root cause?)).

There's a nice proof over at Math Overflow: https://mathoverflow.net/questions/51283/rigourous-proof-of-eulers-identity