You are correct. $F^H = \bar{F}$ since $F^T = F$. Thus $F^{-1} = F^H = \bar{F}$, and so:
$$
\hat{\mathbf{a}} = F\mathbf{a} \Leftrightarrow F^H\hat{\mathbf{a}} = \mathbf{a} \Leftrightarrow \bar{F}\hat{\mathbf{a}} = \mathbf{a}
$$
where $\mathbf{a} = \begin{bmatrix} a_0 & a_1 & \cdots & a_{n-1} \end{bmatrix}^T$ and similar for $\mathbf{\hat{a}}$
Why is $a_k=F^H\hat{a}_k$ and not just $a_k=\bar{F}\hat{a}_k$, i.e. why is here the conjugate-transpose involved and not only the conjugate?
You're notation is a little off here. You have a matrix operating on a scalar. I believe what you intended is the DFT matrix acting on the vector of $a_k$'s as I've written above.
As you can see from above, they are equivalent. So it is perfectly acceptable to write it either way.
Now, the question of why is $F^{-1}$ typically written as $F^H$ instead of $\bar{F}$, is a different question all together. I can't say for certain, but I would conjecture that this is because this is because $F$ is a type of matrix from a larger class, called unitary matrices. For a unitary matrix, $P$, it is the case that:
$$
P^{-1} = P^H
$$
Writing $F^H$ is perhaps more obvious a reminder that the DFT matrix is unitary, and unitary matrices have useful properties which are often used in derivations (such as norm preservation, i.e., Parseval's theorem).
Your definition of incomplete Gamma function is wrong.
As an alternative definition of incomplete Gamma function:
$$\Gamma(v,at)=\int^\infty_1(at)^vu^{v-1}e^{-atu}du=a^vt^v\int^\infty_1u^{v-1}e^{-atu}du$$
Your original integral $I$ becomes
$$I=\int^\infty_0a^vt^v e^{-pt}\int^\infty_1u^{v-1}e^{-atu}du dt$$
By Fubini's theorem (the integrand is always positive),
$$I=a^v\int^\infty_1 u^{v-1} \underbrace{\int^\infty_0 t^ve^{-pt}e^{-atu}dt}_{=I_1} du$$
$$I_1=\int^\infty_0t^ve^{-(p+au)t}dt$$
Let $g=(p+au)t$,
$$I_1=\int^\infty_0\frac{g^v}{(p+au)^{v+1}}e^{-g}dg=\frac{\Gamma(v+1)}{(p+au)^{v+1}}$$
$$I=a^v\int^\infty_1 u^{v-1}\frac{\Gamma(v+1)}{(p+au)^{v+1}}du=a^v\Gamma(v+1)\int^\infty_1\left(\frac{u}{p+au}\right)^{v+1}\frac1{u^2}du$$
Substitute $h=\frac1u$,
$$I=a^v\Gamma(v+1)\int^1_0\frac1{(ph+a)^{v+1}}dh=a^v\Gamma(v+1)\cdot\frac{-1}{pv}\left((p+a)^{-v}-a^{-v}\right)$$
By recognizing $\Gamma(v+1)=v\Gamma(v)$ and further simplifying,
$$I=\frac{\Gamma(v)}p\left(1-\left(1+\frac pa\right)^{-v}\right)$$
An integral transform of Gamma function rarely exist, because Gamma function grows too rapidly. It grows even faster than exponential growth, that's why it does not have a Laplace transform. I'm not sure if a kernel like $e^{-x^2}$ would be okay.
Note that when we try to do a Laplace transform of Gamma function (incomplete or complete) with respect to the first argument, we always fail. If we do it w.r.t. to the second argument (for incomplete Gamma function), we get something useful, which is listed in your table.
Due to $$\lim_{x\to\infty}\frac{\Gamma(x)x^\alpha}{\Gamma(x+\alpha)}=1$$, I can 'invent' a kernel such that the transform for Gamma function exists.
$$\mathcal{T}_\alpha\{f\}(s)=\int^\infty_0 f(t)\cdot\frac{t^{\alpha-s}}{\Gamma(t+\alpha)}dt$$
$\mathcal{T}_\alpha\{\Gamma\}(s)$ exists if one of the following condition is satisfied:
- $\alpha>s>1$
- $\alpha\in\mathbb Z^-\cup\{0\}$ and $s>1$
Best Answer
The real part of the integral is easily obtained thanks to the Fourier transform (see attachment). The imaginary part leads to much more difficulties. We even not know if a closed form exists.
The formula below shows the imaginary part expressed as a Cauchy Principal Value. I doubt that a simpler closed form could be derived.
The numerical tests are in very good agreement with this formula. Sorry, I don't presently publish the analytical calculus leading to the imaginary part because there is still a remaining theoretical difficulty.
[Typo corrected in the formula : 1/pi was missing ]