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