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Rudin 8.6

  • Thread starter ehrenfest
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  • #1
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Homework Statement


Suppose f(x)f(y)=f(x+y) for all real x and y,
(a) Assuming that f is differentiable and not zero, prove that

[tex]f(x) = e^{cx}[/tex]

where c is a constant.

(b) Prove the same thing, assuming only that f is continuous.


Homework Equations





The Attempt at a Solution


(a) The given information implies that f(x)^2 = f(2 x) and furthermore that f(x)^n = f(n x) for any natural number n. We can differentiate that to obtain f(x)^(n-1) f'(x) = f'(n x) for any natural number n.

It is also not hard to show that f'(x) f(y) = f'(y) f(x) for all real x and y.

We can take y = 0 to find that f(0) = 1 (I have used the fact that f is nonzero here).
 

Answers and Replies

  • #2
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For (a), would it not be easier to compare the power series of f(x) and ecx.
 
  • #3
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For (a), would it not be easier to compare the power series of f(x) and ecx.
Perhaps, but I don't know how to prove that f is analytic.
 
  • #4
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You're told f is differentiable which is just another way of saying that it's analytic.
 
  • #5
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differentiability implies analyticity only in complex analysis. Rudin even gives an example of a function that is real-differentiable but not analytic in Exercise 8.1
 
  • #6
lurflurf
Homework Helper
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show that a and b each imply
lim_{h->0}[f(h)-1]/h=c
and then
f(x)f(y)=f(x+y) for all real x and y
with
lim_{h->0}[f(h)-1]/h=c
imply
f(x)=exp(cx)
 
  • #7
2,012
1
show that a and b each imply
lim_{h->0}[f(h)-1]/h=c
and then
f(x)f(y)=f(x+y) for all real x and y
with
lim_{h->0}[f(h)-1]/h=c
imply
f(x)=exp(cx)
I can prove part a). I can see how it would be enough to show that continuity implies lim_{h->0}[f(h)-f(0)]/h exists but I don't know how to do that.
 
  • #8
HallsofIvy
Science Advisor
Homework Helper
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You are given that f is differentiable.

f(x+h)- f(x)= f(x)f(h)- f(x)= f(x)[f(h)- 1].

[tex]f'(x)= \lim_{\stack{h\rightarrow 0}}\frac{f(x+h)- f(x)}{h}= f(x) \lim_{\stack{h\rightarrow 0}}\frac{f(h)- 1}{h}[/itex]
and because f is differentiable, that limit exists.
 
  • #9
2,012
1
You are given that f is differentiable.

f(x+h)- f(x)= f(x)f(h)- f(x)= f(x)[f(h)- 1].

[tex]f'(x)= \lim_{\stack{h\rightarrow 0}}\frac{f(x+h)- f(x)}{h}= f(x) \lim_{\stack{h\rightarrow 0}}\frac{f(h)- 1}{h}[/itex]
and because f is differentiable, that limit exists.
I said I could do part a).

But lurflurf said that (b) also implied that the limit exists.
 

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