The Polar Form for Complex Numbers

Equation (2.13) is not yet in a form that is fundamentally different from the Cartesian form expressed in equation (2.5). However, we can obtain an alternative, more compact, and far more powerful way of writing the polar form of a complex number by re-visiting the Maclaurin series for the sind, cos6 and exponential functions. The Maclaurin series for cosine and sine are  

If we substitute each of these into equation (2.13) we obtain  

The braces in equation (2.17) contain the Maclaurin series for e and so we can rewrite the polar form for z more compactly as  

This important result is known as Euler s formula. 

A The complex conjugate of z can be written in terms of r,Q as above, using the Maclaurin series for cos0 and sin6 , as  

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