Problem evaluating a complicated integral

Question

Apologies for stating my problem poorly in the first instance, thank you for the help in narrowing down the issue.

Problem

For those interested, it's for the exact unbiased inverse of the generalized Anscombe Transform, as given in this paper: http://dx.doi.org/10.1109/TIP.2012.2202675.

I'm trying to integrate, for a given value of $y$ and $\sigma$, the function

$$ \int_{-\infty}^{+\infty} f_{\sigma}\left ( z \right ) \sum_{k=0}^{\infty}\left ( \frac{y^{k}\mathrm{e}^{-y}}{k!\sqrt{2\pi \sigma^{2}}} \mathrm{e}^{-\frac{\left ( z-k \right )^{2}}{2\sigma^{2}}} \right ) \mathrm{d}z $$

where

$$ f_{\sigma}\left ( z \right ) = \left\{\begin{matrix} 2\sqrt{z+\frac{3}{8}+\sigma^{2}}, & z > -\frac{3}{8}-\sigma^{2}\\ 0, & z \leq -\frac{3}{8}-\sigma^{2} \end{matrix}\right. $$

which gives the equation in the reference below:

$$ \int_{-\infty}^{+\infty} 2\sqrt{z+\frac{3}{8}+\sigma ^{2}}\sum_{k=0}^{\infty}\left ( \frac{y^{k}\mathrm{e}^{-y}}{k!\sqrt{2\pi \sigma^{2}}} \mathrm{e}^{-\frac{\left ( z-k \right )^{2}}{2\sigma^{2}}} \right ) \mathrm{d}z $$

In the reference, the values of the above function are tabulated, presumably in Matlab, for $\sigma \in \left \{ 0.01,...,50 \right \}$ and $y \in \left \{0,...,200 \right \}$. The authors have made this table available for download, but I want to do it myself.

However, I can't get it to work - Mathematica won't evaluate it:

func[z_, y_, sig_] := 
 Piecewise[{{2 Sqrt[z + 3/8 + sig^2],z > -3/8 - sig^2}},0.]*
  Sum[((y^k )* Exp[-y])/(k! * Sqrt[2 \[Pi] sig^2]) * 
    Exp[-((z - k)^2) / (2 sig^2)], {k, 0, Infinity}]
NIntegrate[func[z, 5, 2], {z, -Infinity, Infinity}]

Have I made a mistake somewhere? Or misunderstood the integral/problem as given?

Update

A suggestion is to replace the infinite sum with an upper limit of ~50 as it converges quickly. This does allow Mathematica to now evaluate the integral.

Answer 1

It cannot be real, since zunder the radical goes to minus infinity. Anyway, if you only need a numerical table, why do not you do something like this:

    f[y_, sig_, n_] := 
 NIntegrate[
  2 Sqrt[z + 3/8 + sig^2]*
   Sum[((y^k)*Exp[-y])/(k!*Sqrt[2 \[Pi] sig^2])*
     Exp[-((z - k)^2)/(2 sig^2)], {k, 0, 
     n}], {z, -\[Infinity], \[Infinity]}]

Here nis the number of terms in the sum. It converges rather fast. To check it let us do the following:

 f[5, 2, #] & /@ {10, 100, 1000}

(*   {5.92884 + 0.000418281 I, 6.03812 + 0.000418544 I, 
 6.03812 + 0.000418544 I}   *)

The result is complex, which is to be expected, and not related to the number of terms in the sum.

Answer 2

Assuming that if the integrand is complex we should take it to be zero, you can do this:

uplim = 50;
arg = If[Im@# > 0, 0, #] &@
   N[2 Sqrt[z + 3/8 + 2^2]*
     Total@Table[((5^k)*Exp[-5])/(k!*Sqrt[2 \[Pi] 2^2])*
        Exp[-((z - k)^2)/(2 2^2)], {k, 0, uplim}]
    ];
Plot[arg, {z, -10, 20}]
NIntegrate[arg, {z, -\[Infinity], \[Infinity]}]

This constructs the series by hand, up to some upper value (I worked out that 50 is enough by looking at the plot