09/09/2014, 04:33 AM
(This post was last modified: 09/09/2014, 01:05 PM by sheldonison.)

(09/08/2014, 11:03 AM)tommy1729 Wrote: I looked at sheldon's 2nd answer and noticed what partially was the motivation for one of my last questions :

the plot of fake ln(x) resembles integral (1 - exp(-x) ) / x !

It was "Visually Obvious" from the psuedoperiodicity and the growth rate for Re(z) << 0.

I havent considered the signs of the derivatives of integral (1 - exp(-x) / x yet.

Maybe that explains alot.

regards

tommy1729

I reread your post. So, to clarify, you are interested in comparing the fake function for with the integral (which I don't yet understand), integral (1 - exp(-x) ) / x.

So far, I have only answered the Op's question, and generated the "fake" function for but doing so for or or or would presumably be straightforward enough using the equation I posted in my 2nd answer; copied below.

fake function for exp(z)ln(z+1), replace ln(1+e^{}) with other positive valued function

the fake(z) zeros are probably always near the imaginary axis apart. Also, I posted the first 18 zeros of the asymptotic function for at MSE, along with an approximation.

It would be interesting if this equation is related to the integral solution posted on Mathoverflow, but I don't understand that solution (yet).

- Sheldon