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Conjecture and prove a statement about a differentiable function satisfying f(x+a) = bf(x)

Let f(x) be a function which is differentiable everywhere and which satisfies

    \[ f(x+a) = bf(x) \]

for some positive constants a and b. What can you conclude about such a function f?


(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 f(x) satisfies the functional equation f(x+a) = bf(x) we can write

    \[ f(x) = b^{\frac{x}{a}} g(x) \]

which implies

    \[ g(x) = \left( \frac{1}{b} \right)^{\frac{x}{a}} f(x). \]

Then computing

    \begin{align*}   g(x+a) &= \left( \frac{1}{b} \right)^{\frac{x+a}{a}} f(x+a) \\  &= \left(\frac{1}{b}\right)^{\frac{x}{a}} \cdot \frac{1}{b} \cdot bf(x) \\  &= \left( \frac{1}{b}  \right)^{\frac{x}{a}} f(x) \\  &= g(x). \end{align*}

Thus, g(x) is indeed periodic with period a and so

    \begin{align*}  f(x+a) &= b^{\frac{x+a}{a}} g(x+a) \\  &= b\cdot b^{\frac{x}{a}} g(x) \\  &= b f(x). \end{align*}

So, this definition of f(x) in terms of the periodic function g(x) indeed satisfies the functional equation.

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