I have data measuring an exponential decay that is convoluted by a gaussian response function.
I have the measured shape of the gaussian, and want an analytical expression for the exponential post-convolution that I can use to compare to the data.
I need to calculate the following, but am having trouble.
$g(\tau) = \int_-^\infty \exp(-\lambda t) \exp(-\frac{(t-\tau)^2}{2\sigma^2} ) d \tau$
Where $\sigma$ is known.
$g(\tau) = \int_-^\infty \exp(-\lambda t -\frac{t^2}{2\sigma^2} +\frac{t \tau}{\sigma^2}) \exp(-\frac{\tau^2}{2\sigma^2} ) d \tau$
The last term looks like the Error function, but Im not sure is it.
Note that the algebraic identity $$\lambda t+\frac{(t-\tau)^2}{2\sigma^2}=\tau\lambda-\frac12\sigma^2\lambda^2+\frac{(t-\tau+\sigma^2\lambda)^2}{2\sigma^2} $$ and the change of variable $s=t-\tau+\sigma^2\lambda$ yield $$ \int_{-\infty}^{\infty}\exp\left(-\lambda t\right)\,\exp\left(-\frac{(t-\tau)^2}{2\sigma^2}\right)\mathrm dt=\exp\left(-\tau\lambda+\frac12\sigma^2\lambda^2\right)\cdot\int_{-\infty}^{\infty}\exp\left(-\frac{s^2}{2\sigma^2}\right)\mathrm ds, $$ that is, $$ g(\tau)=\sqrt{2\pi\sigma^2}\cdot\exp\left(-\tau\lambda+\frac12\sigma^2\lambda^2\right). $$ This assumes that the function $g$ is defined as $$ g(\tau) = \int_{-\infty}^\infty \exp(-\lambda t) \exp\left(-\frac{(t-\tau)^2}{2\sigma^2} \right)\mathrm d t, $$ since the current formula in the question makes no sense (subscript $-$ in the integral, presumably instead of $-\infty$, $\mathrm d\tau$ to integrate a function of $t$, presumably instead of $\mathrm dt$).