Problem statement: Given the complex numbers z and w, determine:
the machine and Mathcad yield:
so we know what the answer should be.
We are given in literature:
But this is NO FUN!
Lets use the theorems, laws, and identities we know to develop our own solution.
Solving zc using de Moivre's identity:
(where c is a real number)
Solving zdi using Eulers formula: (twice)
simplifying each of these two parts individually:
what to do now? From http://en.wikipedia.org/wiki/Exponentiation#Summary,:
do you understand this from logarithms!
using Eulers formula (again):
so putting it all together:
Just for kicks, lets have the machine confirm our solution:
which is the same as the machine when it computes everything itself:
by observation we see the phasor form for z:
Complex numbers are easily added, subtracted, multiplied, and divided
For complex numbers to a real power (n), we use de Moivre's identity:
(for real values of n, NOT when n is complex)
Also study next page 299 to know Euler's Formula:
For complex exponentiation (a complex number to a complex exponent), know the derivation of the following:
Euler's Formula derivation/proof using Taylor Series expansions (one of many ways) of ex, sine and cosine:
basic facts about the powers of i:
For complex z we define each of these functions by the above series, replacing x with z (an imaginery number). We find that: