I am interested in finding a list of integrals of the form:
$$ \int a^n e^{iax} da$$
For $n = 0$, I found it to be $\delta(x)$.
For $n = -1$, I found it to be sgn$(x)$.
For $n = -2$, I found it to be $\lvert x \rvert $.
But what is the value for $n = 1$? And more generally, is there a table of integrals evaluating the above for any integer $n$? These appear to be widely used integrals yet I was unable to find their exact expressions documented in tables.
In the classical sense, the integrals do not exist for any $n$, at the value $x=0$. For $n\geq 0$, this is because $a^n$ does not decay at infinity. For $n<0$, the singularity at $0$ is too poor. If we instead consider the tempered distributions associated with $a^n$, that is the functional $T_{a^n}\in \mathcal{S}'(\mathbb{R})$ given by $T_{a^n} (\phi) = \int_\mathbb{R} a^n \phi\, da$ against test functions $\phi \in\mathcal{S}(\mathbb{R})$, then the Fourier transforms exist as distributions. The Fourier transform (both the normal one and the one on distributions) satisfies $\mathcal{F}(\frac{df}{da})=ix\mathcal{F}(f)$. For $n>0$, this gives us $\mathcal{F}(T_{a^n}) = \frac{1}{i^n}\delta^{(n)}$, where $\delta^{(n)}$ is the nth distributional derivative of $\delta$. For negative powers, if $n>0$, $\mathcal{F}(T_{a^{-n}})$ is the distribution associated with $\frac{(ix)^n}{|x|}$