Determine all values for the real numbers and such that the following equations hold.

- .
- .
- .
- .
- .
- .

- The equation
The value of is arbitrary.

- Using the formula for the absolute value of a complex number we have
Since the equation implies which implies . Therefore, this equation is satisfied by

(

**Note:**The answer Apostol gives says , but I think works as well. - Again, using the formula for the absolute value of a complex number we have
This holds for all real and .

- We compute as follows,
Hence, we must have either and is arbitrary or arbitrary and .

- We compute,
This gives us two equations (since the real parts and imaginary parts must be equal),

If then from the second equation we have

If then we have so or . But, is not impossible since then is undefined. Therefore we have two possibilities

(

**Note:**Apostol only gives the first of these solutions. We can check by a direct substitution that the second solution also works.) - Here we note that
Therefore, we have

Therefore, from the given equation we have