Owen's T statistical function —> Taylor development?

[Gil]

I have got the Owen's T statistical function
'1/(2*pi)*Integral (from 0,to a,e^(-h^2*((1+t^2)/2))/(1+t^2),t) '.

Normally, h & a are real numbers.

Wolfram Alpha gives an example with a complex argument.

I imagine that, to achieve that, the original function with the integral was transformed with a Taylor's development or some alike idea.

For real h=4 and a=1, the numerical integral on HP50 gives correctly 1.58351193827E-5 (as in Wolfram).

I tried the Taylor development with the following program, but I get alternatively a positive and a negative value, but no sign of converging value toward 1.58351193827E-5:

\<< 4 1 \-> h a
\<< '1/(2*\pi)*e^(-h^2*((1+t^2)/2))/(1+t^2)' EVAL DUP 'f' STO 'last.deriv' STO f 't=0' SUBST EVAL a * \->NUM 2 40
FOR i last.deriv 't' \.d EVAL DUP 'last.deriv' STO 't=0' SUBST EVAL a i ^ * i ! / \->NUM +
NEXT
\>>
\>>

Could somebody try and help me give the searched approached "solution" ?

Apparently not an easy task, after discovering:

https://www.researchgate.net/publication...T_Function

And, supposing that the found program or Taylor's algorithm is OK for real number, can we use it with complex numbers?

Many thanks in advance.