Conditions for which two functions, one containing an integral, are equal

Question

Define two functions:

${\beta _1}(\tau ,\omega ) = \exp \left[ {\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} } \right]$

${\beta _2}(\tau ,\omega ) = \exp \left[ {\frac{{\omega \tau }}{{2Q(\tau )}}} \right]$

All variables in the above are real numbers, and $Q(\tau)$ is a real-valued function.

For what conditions are the two functions equal, such that $\beta_1(\tau, \omega) = \beta_2(\tau, \omega)$?

What can be said about equality between the two functions over an interval $[\tau_1,\tau_2]$, when the function $Q(\tau)$ is approximately constant over this interval?

Answer 1

For the functions in your questions to be equal, we must have

$\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} = \frac{{\omega \tau }}{{2Q(\tau )}}$

Taking the derivative of both sides with respect to $\tau$ we get

$\frac{\omega }{{2Q(\tau)}} = \frac{{\omega Q(\tau ) - \omega \tau Q'(\tau )}}{{2(Q(\tau ))^2}}$

Which must hold for any $\omega,\tau$.

Hence, following should be true $\frac{{Q(\tau ) - \tau Q'(\tau )}}{{Q(\tau )}} = 1$. This is only possible when $\tau Q'(\tau ) = 0$. Since $Q'(\tau ) \neq 0 $, for $\forall \tau$ the only valid value of $\tau$ is 0.