I know $\exp(z) , \sin(z) , \sin^2(z) , z \exp(z) , \sinh(z) , \exp(z) \sin(z) $ all have addition formulas.

I'm still confused about addition formulas.

It seems hard to tell if a function has an addition formula or not.

Perhaps related : About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$

Analytic solutions to $f(x+y) +h(x+y)= f(x)(g(y)+h(y)) + g(x)(f(y)+h(y)) + h(x)(f(y)+g(y))$?

Visual proof of the addition formula for $\sin^2(a+b)$?

Second question :

Does every function with an addition formula grow exponentially fast ?

  • $\begingroup$ For your second question, it depends on how rigidly you define an addition formula; $f(x)=0$ satisfies $f(x+y)=f(x)+f(y)$ (amongst other possible formulae). The answer is probably no, though. $\endgroup$ – πr8 Dec 31 '16 at 12:24
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    $\begingroup$ @πr8 Using that addition formula, $f(x)=ax$ also works and does not grow exponentially. $\endgroup$ – Michael Burr Dec 31 '16 at 12:38
  • $\begingroup$ It is hard, in general, to determine if a function has an addition formula (or satisfies any functional relationship). $\endgroup$ – Michael Burr Dec 31 '16 at 12:39
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    $\begingroup$ It is a famous, though unfashionable, theorem of Weierstrass that any meromorphic function with an algebraic addition theorem (i.e. $G(f(u),f(v),f(u+v))=0$ for a polynomial in 3 variables $G$) if and only if it is one of the following: 1) a rational function of $z$, 2) a rational function of $e^{\lambda z}$ for some $\lambda$, or 3) an elliptic function. See here §2 for a proof, although the author's style is quite enthusiastic in places. $\endgroup$ – Chappers Dec 31 '16 at 12:59
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    $\begingroup$ For the second part, note that, for example, $f(x)=x^2$ has the addition formula $(f(a+b)-f(a)-f(b))^2=4f(a)f(b)$, and does not grow exponentially. $\endgroup$ – Chappers Jan 2 '17 at 18:04

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