Best Answer

Euler's formula states that eix = cos(x) + isin(x) where i is the imaginary number and x is any real number.

First, we get the power series of eix using the formula:

ez = Î£âˆžn=o zn/n! where z = ix. That gives us:

1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7! + (ix)8/8! + ...

which from the properties of i equals:

1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! - x6/6! - ix7/7! + x8/8! + ...

which equals:

(1 - x2/2! + x4/4! - x6/6! + x8/8! - ...) + i(x - x3/3! + x5/5! - x7/7! + ...).

These two expressions are equivalent to the Taylor series of cos(x) and sin(x). So, plugging those functions into the expression gives cos(x) + isin(x).

Q.E.D.

Of course, were you to make x = Ð¿ in Euler's formula, you'd get Euler's identity:

eiÏ€ = -1

It depends on your definitionThe question presumes that someone has already given a definition of the exponential function for complex numbers. But has this definition been given? If so, what is this official definition?The answer above assumes that the exponential function is defined, for all complex numbers, by its power series. (Or, at least, that someone else has already proved that the power series definition is equivalent to whatever we're taking to be the official definition).

So the answer depends critically on what the definition of the exponential function for complex numbers is.

Suppose you know everything about the real numbers, and you're trying to build up a theory of complex numbers. Most people probably view it this way: Mathematical objects such as complex numbers are out there somewhere, and we have to find them and work out their properties. But mathematicians look at it slightly differently. If we can construct a thing which has all the properties we feel the field of complex numbers should have, then for all practical purposes the field of complex numbers exists. If we can define a function on that field which has all the properties we think exponentiation on the complex numbers should have, then that's as good as proving that complex numbers have exponentials. Mathematicians are comfortable with weird things like complex numbers, not because they have proved that they exist as such, but because they have proved that their existence is consistent with everything else. (Unless everything else is inconsistent, which would be a real pain but is very unlikely.)

So how do we construct this exponential function? One approach would be simply to define it by exp(x+iy) = ex(cos(y)+i.sin(y)). Then the answer to this question would be trivial: exp(iy) = exp(0+iy) = e0(cos(y)+i.sin(y)) = cos(y)+.sin(y). But you'd still have some work to do to prove things like exp(z+w) = exp(z).exp(w). Alternatively, you could define the exponential function by its power series. (There's a theorem that lets you calculate the radius of convergence, and that tells us the radius is infinite, i.e. the power series works everywhere.) Or maybe you could try something like proving that the equation dw/dz = w has a unique solution up to a multiplicative constant, and defining the exponential function to be the solution which satisfies w=1 at z=0.

Let z = cos(x) + i*sin(x)

dz/dx = -sin(x) + i*cos(x)

= i*(i*sin(x) + cos(x))

= i*(cos(x) + i*sin(x))

= i*(z)

therefore dz/dx = iz

(1/z)dz = i dx

INT((1/z)dz = INT(i dx)

ln(z) = ix+c

z = e(ix+c)

Substituting x=0, we get cos(0) + i*sin(0) = e(0i+c)

1+0 = e(0+c)

e(0+c) = 1

therefore c=0

z = eix

eix = cos(x) + i*sin(x)

Substituting x=pi, we get e(i*pi) = -1 + 0

e(i*pi) = -1

e(i*pi) + 1 = 0

qED

Q: How do you prove Euler's formula?

Write your answer...

Submit

Still have questions?

Continue Learning about Math & Arithmetic

Euler published the formula, which relates complex exponentials to trigonometric functions in 1748. See related link.

Eulers number Approx x^2.31

You would first prove that 3 of the sides are the same size (The 4th is automatically the same size as the other three). Then, you would prove that there are three right angles in the shape, using the slopes (If lines make a right angle, they are perpendicular and their slopes are negative reciprocals of each other.).

You really can't "prove" the formula. You use it. You first square the base 'b'. Then, you multiply that number by the height 'h'. Then, you divide the product of the base squared and height by 3. Boom! You get your answer. In my school, we get a formula sheet with all the formulas we will need to use. If you didn't understand the description above, here is the formula for a square pyramid:1/3b2 h.Hope this helped!

I think the following Wikipedia link on Fourier Series (see related links below), has the information that you're looking for.

Related questions

Euler published the formula, which relates complex exponentials to trigonometric functions in 1748. See related link.

He discovered the all important Euler's Rule often referred to as Euler's Formula.

Kathrin Eulers has written: 'Frauen im Wahlrecht'

cool

no

http://en.wikipedia.org/wiki/Euler_angles

why is eulers constant important

The formula for the area of a rectangle is length x breadth. In order to prove this works, work out an area of a rectangle using that formula.

It's about ponis and viagra.

Eulers number Approx x^2.31

Prove it by induction on n, use 0 or 1 as base cases.

It is given that name because of eulers work with e