Let be a function which is differentiable everywhere and which satisfies
for some positive constants and . What can you conclude about such a function ?
(Note: I’m not entirely sure what Apostol wants here since the instruction “what can you conclude” is pretty vague. He does give an “answer” in the back of the book, so I verify that it does have the properties indicated, but I don’t know how you would arrive at that expression just from the question statement. I’ll mark this question as incompletely and hopefully come up with something better in the future.)
Since satisfies the functional equation we can write
Thus, is indeed periodic with period and so
So, this definition of in terms of the periodic function indeed satisfies the functional equation.