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Determine all real numbers satisfying given relations

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

1. .
2. .
3. .
4. .
5. .
6. .

1. The equation

The value of is arbitrary.

2. 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.

3. Again, using the formula for the absolute value of a complex number we have

This holds for all real and .

4. We compute as follows,

Hence, we must have either and is arbitrary or arbitrary and .

5. 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.)

6. Here we note that

Therefore, we have

Therefore, from the given equation we have