# Function as a Solution to Specific Conditions

• I
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
Mentor
##f(x) = Ce^{\pm x}##

##f(x) = Ce^{\pm x}##
Nice. How did you work that out? Is an exponential function the only solution to this problem?

DrClaude
Mentor
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.

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
Mentor
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.

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.

mfb
Mentor
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