# Proving two functions are equal under an integral

1. Dec 14, 2009

### dreNL

I have the following problem.
If
$$\int_0^\infty f(s)ds=\int_0^\infty g(s)ds$$
What are sufficient conditions such that $$f(s)=g(s)$$?

I know that two functions $$f(s),g(s)$$ are equal if their domain, call it $$S$$, is equal and if $$f(s)=g(s)$$ for all $$s\in S$$ but I can't figure this one out.

The full problem is actually

$$-\int_0^\infty\frac{d}{ds}\left(f(\vec{r}+s\hat{\Omega},E,\hat{\Omega})e^{-\Sigma_ts}\right)ds$$
$$=\int_0^\infty\Sigma_te^{-\Sigma_ts}g(\vec{r}+s\hat{\Omega},E,\hat{\Omega})ds$$
and therefore hopefully
$$-\frac{d}{ds}\left(f(\vec{r}+s\hat{\Omega},E,\hat{\Omega})e^{-\Sigma_ts}\right)=\Sigma_te^{-\Sigma_ts}g(\vec{r}+s\hat{\Omega},E,\hat{\Omega})$$

Please help me!!! I'm finishing soon with my work and proving this (or at least having sufficient conditions) would be very welcome!

2. Dec 14, 2009

### HallsofIvy

Just knowing that the integrals over a specified interval are 0 tells you essentially nothing about f and g themselves. The only "sufficient" condition is that f(x)= g(x)!

3. Dec 18, 2009

### dreNL

Thank you HallsofIvy,

How about if I restate the problem as:
$$\int_0^\infty e^{-\Sigma_t s}f(s)ds=\int_0^\infty e^{-\Sigma_t s}g(s)ds$$
And it should hold for any $$\Sigma_t \in [0,\infty)$$

for very large $$\Sigma_t\to\infty$$, the only contribution is just right from zero and because the integrations over a very small interval return the same value I can imagine $$f(s)=g(s)$$ there. For slightly less large $$\Sigma_t$$ the integrations still return the same value and so forth.
This is by no means mathematical proof (as I am not a mathematician). From a physical point of view it makes some sense..
Actually, does it help if I tell you the functions $$f(s),g(s)$$ I am considering are all well behaved (finite number of discontinuities)?

As you told me I need to proove that $$f(s)=g(s)$$ for every any $$s\in [0,\infty)$$. Is there now any hope of stating conditions for $$f(s),g(s)$$ so that I can use the above to make it plausible that $$f(s)=g(s)$$ or even give proof?