I am trying to calculate the following integral, $$ \int_{0}^\infty dk e^{i(\alpha k^2 + \beta k +\gamma)} $$ where $\alpha,\beta,\gamma$ are real numbers and $\alpha>0$. I am aware that the integral does not converge but, when the exponent is a polynomial of degree $1$, a formal expression can be obtained by using the Sokhotski-Plemelj identity $$ \int_{0}^\infty dk e^{isk} = \pi\delta(s) +i \mathcal{P}\frac{1}{s}, $$ with $\delta(x)$ the Dirac delta distribution and $\mathcal{P}$ representing the Cauchy principal value. I am interested in calculating the first integral in these terms, but I don´t manage to find a generalization (if any) of the above rule. Does anybody know whether there is an expression of the above integral in terms of a Dirac Delta and a principal value?
Thanks!
The integral does converge. The change of variables $$k = \frac{s}{\sqrt{\alpha}} - \frac{\beta}{2\alpha} $$ gives you
$$ \frac{\exp(i \gamma - i \beta^2/(4\alpha))}{\sqrt{\alpha} }\int_{\beta/(2\sqrt{\alpha})}^\infty \exp(i s^2) \; ds $$
It is known (see Fresnel integral) that $$ \int_0^\infty \exp(is^2)\; ds = (1+i) \sqrt{\frac{\pi}{8}}$$