better understanding of incomplete gamma function $\Gamma(0,x)$ By definition  incomplete Gamma function is:$$\Gamma(0,x)=\int_{x}^{\infty}t^{-1}e^{-t}dt $$
I have an expression which includes  
$$\Gamma(0,r(A)e^{i\phi(A)}),$$ where $A>0$ is a parameter, and $r(A)$ is almost linear, $\displaystyle \phi(A)\rightarrow\pm\frac{\pi}{2},\ as\ A\rightarrow\infty  $  
I want to have more intuition about the influence of parameter $A$. 
I will be very appreciate any comments or a good reference.
 A: When you say $r(A)$ is almost linear, do you mean that for large $A$ that $r(A)\approx\rho A+c$ for some constants $\rho$ and $c$?  Also, what do you mean by $\phi(A)\rightarrow\pm\frac{\pi}{2}$?  
Without the $\pm$, and with the above assumption about $r(A)$:
$$\frac{d}{dA}\Gamma(0,r(A)e^{i\phi(A)})=-\left[r(A)e^{i\phi(A)}\right]^{-1}e^{-\left[r(A)e^{i\phi(A)}\right]}\left(r'(A)e^{i\phi(A)}+r(A)i\phi'(A)e^{i\phi(A)}\right)$$
For large $A$, $r(A)\approx\rho A+c$, $r'(A)\approx\rho$, $\phi(A)\approx\frac{\pi}{2}$, and $\phi'(A)\approx0$.  And the above reduces to:
$$\frac{d}{dA}\Gamma(0,r(A)e^{i\phi(A)})\approx-\left[(\rho A+c) i\right]^{-1}e^{-\left[(\rho A+c)i\right]}\left(\rho i +0\right)$$
$$\frac{d}{dA}\Gamma(0,r(A)e^{i\phi(A)})\approx-\frac{1}{A+\frac{c}{\rho}}e^{-(\rho A+c)i}$$
So for large $A$, increasing $A$ by a differential $\Delta A$ will "add" about $\frac{\Delta A}{A+\frac{c}{\rho}}$ to your quantity, but in the direction of $-e^{-(\rho A+c)i}$; that is it will add a complex differential of about $\frac{\Delta A}{A+\frac{c}{\rho}}$ in magnitude at an angle of $-\rho A-c+\pi$. 
Imagine an angle that spins around the origin (clockwise if $\rho$ is positive) at a linear rate with respect to $A$.  Now add to your quantity in that rotating direction with a magnitudes that decrease roughly like $\frac{1}{A}$.
A: The incomplete $\Gamma$ function behaves like
$$\Gamma(0,z) \sim \frac{e^{-z}}{z} \, \sum_{k=0} \frac{(-1)^k k!} {z^{k}},$$
valid for $|\text{arg} z| < 3\pi/2$, see here.
If you substitute $$z = r(A) e^{i\phi(A)}$$ and take a few terms (maybe even only one), you will know how $\Gamma(0,z)$ behaves as a function of $A$.
A: I've done an (amateurish) discussion on the gamma-function some monthes ago. I arrived at the incomplete gamma by some naive approach. Possibly this gives some additional intuition/insight to you. See "uncompleting the gamma"
