Problem statement: Given the complex numbers z and w, determine:

Given:

we know:

Given (in phasor form):

we know:

since:

the machine and Mathcad yield:

so we know what the answer should be.

We are given in literature:

so substituting...

But this is NO FUN!

Lets use the theorems, laws, and identities we know to develop our own solution.

Given:

(1st law of exponents)

Breaking

into two parts:

and

Solving zc using de Moivre's identity:

(where c is a real number)

remembering:

substituting:

Solving zdi using Eulers formula: (twice)

Eulers formula:

do you understand this!

(3rd law of exponents)

simplifying each of these two parts individually:

part 1

(3rd law of exponents)

what to do now? From http://en.wikipedia.org/wiki/Exponentiation#Summary,:

part 2

do you understand this from logarithms!

using Eulers formula (again):

so putting it all together:

(answer)

Just for kicks, lets have the machine confirm our solution:

Remembering:

substituting yields:

which is the same as the machine when it computes everything itself:

Remember the following:

(rectangular form)

(the complex conjugate)

by observation we see the phasor form for z:

the real part:

imaginary part:

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:

Let:

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:

etc.

For complex z we define each of these functions by the above series, replacing x with z (an imaginery number). We find that: