Function as a Solution to Specific Conditions

[tade]

I'm trying to solve some statistical mechanics. This problem appeared.

$\frac{f\left(0\right)}{f\left(a\right)}f\left(x+a\right)=f\left(x\right)$

Any idea as to which function will satisfy this equation?

 

[DrClaude]

$f(x) = Ce^{\pm x}$

 

[tade]

$f(x) = Ce^{\pm x}$

Nice. How did you work that out? Is an exponential function the only solution to this problem?

 

[DrClaude]

Nice. How did you work that out?

From the observation that $f(x) \propto f(x+a) / f(a)$ requires that an addition in the argument becomes an (inverse) multiplication of the function, and I recognized the exponential. (I actually thought about log first, but then realized I had it backwards )

Is an exponential function the only solution to this problem?

No idea. Apart from the answer I gave and $f(x) = \mathrm{const.}$, I don't see any.

 

[tade]

From the observation that $f(x) \propto f(x+a) / f(a)$ requires that an addition in the argument becomes an (inverse) multiplication of the function, and I recognized the exponential. (I actually thought about log first, but then realized I had it backwards )

That's pretty neat. What branch of mathematics is that under?

 

[DrClaude]

That's pretty neat. What branch of mathematics is that under?

I'm not a mathematician. It just comes from years of working with, and getting a feeling for, those kind of mathematical functions.

 

[tade]

I'm not a mathematician. It just comes from years of working with, and getting a feeling for, those kind of mathematical functions.

No, I was just wondering what topic or branch this question falls under.

 

[jedishrfu]

My guess is you'd learn this in Calculus especially when studying limits and using Taylor series to approximate a function:

http://mathworld.wolfram.com/TaylorSeries.html

 

[mfb]

The general solution is $f(x) = Ce^{d x}$, C and d arbitrary constants. C and d can be complex if you work with complex numbers.
It is a typical homework problem to show this.

 

[Stephen Tashi]

No, I was just wondering what topic or branch this question falls under.

"Functional equations" https://en.wikipedia.org/wiki/Functional_equation