Ratio of complex valued incomplete gamma function

Question

I've been stuck for a long time with this problem, but what is the value of the following: $\lim_{z\rightarrow \infty}\dfrac{\Gamma(k,iz)}{\Gamma(k)}$, where $z$ is real?

Thanks!

Edit: clarification on z

Answer 1

So you are looking for $$ \mathop {\lim }\limits_{y\; \to \;\infty } {{\Gamma \left( {s,iy} \right)} \over {\Gamma \left( s \right)}} = \mathop {\lim }\limits_{y\; \to \;\infty } Q\left( {s,iy} \right)\quad \left| {\;y \in R} \right. $$ (and we do not specify, for the moment, the field for $s$).

It is known that the Regularized Incomplete Gamma has the following representation $$ Q\left( {s,z} \right) = 1 - {{z^{\,s} \;e^{\,\, - z} } \over {\Gamma \left( s \right)}}\sum\limits_{0\, \le \,k} {{{z^{\,k} } \over {s^{\,\overline {\,k + 1\,} } \,}}} $$ which is tied to the power series expansion of the Lower Gamma function $\gamma(s,z)$ which is holomorphic in $z$ and $s$.

The expansion at $z = \infty$ comes then to be $$ Q\left( {s,z} \right) \approx {{z^{\,s - 1} \;e^{\,\, - z} } \over {\Gamma \left( s \right)}} \left( {1 - {{1 - s} \over z} + O\left( {{1 \over {z^{\,2} }}} \right)} \right)\quad \left| {\;z\, \to \;\infty } \right. $$

Therefore in our case $$ Q\left( {s,iy} \right) \approx {{\left( {iy} \right)^{\,s - 1} \;e^{\,\, - iy} } \over {\Gamma \left( s \right)}} $$ which means that the phase is undefined, while the modulus is asymptotic to $$ \left| {Q\left( {s,iy} \right)} \right| \approx {{\left| y \right|^{\,s - 1} } \over {\Gamma \left( s \right)}} $$ and thus depends on the value of $s$