Double integral $\int_a^b \int_a^y e^{y^2-x^2}dxdy$

Question

I am trying to find a numerical way to compute the following double integral \begin{equation} \int_a^b \int_a^y e^{y^2-x^2}dxdy \end{equation}

One attempt I've made is to express the inner integral in terms of the error-function \begin{equation} \int_a^y e^{-x^2}dx = \frac{\sqrt{\pi}}{2}(\operatorname{erf}y - \operatorname{erf}a) \end{equation} but found no nice way to compute integrals of $e^{y^2}\operatorname{erf}y$.

Answer 1

Let x=r$cost$ and y=r$sint$ then $y^2-x^2$=$-r^2cos2t$ so our integral transforms to ;

$\iint e^{y^2-x^2} dxdy$=$\iint e^{-r^2cos2t} r dr dt$

which can be computed with u-substiution.Let $r^2$=u then

$\int e^{-r^2cos2t} r dr = 1/2 \int e^{-ucos2t} du$ =$\frac{1}{(-2cos2t)} e^{r^2}$

=$\iint e^{-r^2cos2t} r dr dt$=$\int \frac{1}{(-2cos2t)} e^{r^2} dt $

which is just integral of sec