# Proof of Euler's formula

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### Using Taylor series

Here is a proof of Euler's formula using Taylor series expansions
as well as basic facts about the powers of *i*:

and so on. The functions *e*^{x}, cos(*x*) and sin(*x*) (assuming *x* is real) can be written as:

and for complex *z* we *define* each of these function by the above series, replacing *x* with *iz*. This is possible because the radius of convergence of each series is infinite. We then find that

The rearrangement of terms is justified because each series is absolutely convergent. Taking *z* = *x* to be a real number gives the original identity as Euler discovered it.

### Using calculus

Define the function by

This is allowed since the equation

implies that is never zero.

The derivative of is, according to the quotient rule:

Therefore, must be a constant function. Thus,

### Using ordinary differential equations

Define the function by

Considering that is constant, the first and second derivatives of are

because by definition. From this the following 2^{nd} order linear ordinary differential equation is constructed:

or

Being a 2^{nd} order differential equation, there are two linearly independent solutions that satisfy it:

Both and are real functions in which the 2^{nd} derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is

for any constants and But not all values of these two constants satisfy the known initial conditions for :

- .

However these same initial conditions (applied to the general solution) are

resulting in

and, finally,