Since is the algebraic expression of in terms of its rectangular coordinates, the corresponding expression in terms of its polar coordinates is
There is another, more powerful representation of in terms of its polar coordinates. In order to define it, we must introduce Euler's identity:
We'll call the polar form of the complex number , in contrast with the rectangular form . Polar form often simplifies algebraic manipulations of complex numbers, especially when they are multiplied together. Simple rules of exponents can often be used in place of messier trigonometric identities. In the case of two complex numbers being multiplied, we have
A corollary of Euler's identity is obtained by setting to get
This has been called the ``most beautiful formula in mathematics'' due to the extremely simple form in which the fundamental constants , and 0 , together with the elementary operations of addition, multiplication, exponentiation, and equality, all appear exactly once.
For another example of manipulating the polar form of a complex number, let's again verify , as we did above in Eq. (2.4), but this time using polar form:
As mentioned in §2.7, any complex expression can be conjugated by replacing by wherever it occurs. This implies , as used above. The same result can be obtained by using Euler's identity to expand into and negating the imaginary part to obtain , where we used also the fact that cosine is an even function ( ) while sine is odd ( ).
We can now easily add a fourth line to that set of examples:
Thus, for every .
Euler's identity can be used to derive formulas for sine and cosine in terms of :
Similarly, , and we obtain the following classic identities: