Searching for an asymptotic to exp[0.5]

[sheldonison]

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

...
So far, I have only answered the Op's question, and generated the "fake" function for \( \ln(z+1) \approx \exp(-z)\text{fakelnx}(\exp(z)\ln(z+1))
\) but doing so for \( \ln(x) \) would presumably be straightforward enough

Hey Tommy,
Actually, its much more elegant to generate the fake function for ln(x) (than ln(x+1)), and, as you correctly guessed, it is exactly the same as the integral of (1 - exp(-x) ) / x solution! Of course, I can't prove it (yet), but numerically, its exact to the limit that I can generate the fake function Taylor series. The "fakelnx" solution equation simplifies for ln(x) to:

\( a_n =\lim_{r\to\infty} \int_{-\pi}^{\pi} \frac{1}{2\pi}e^{-n(r+ix)}\exp(e^{r+ix})(r+ix)\; \mathrm{d}x\;\; \) for large n, the integral at r=ln(n) works extremely well
\( \text{fakelnx}(z) = \exp(-z) \, \sum_{n=0}^{\infty} a_n x^n\;\; \)

The integral solution is a simple entire Taylor series, though it does require twice as many terms to get good convergence. Do you have any links to the integral of integral of (1 - exp(-x) ) / x solution to the asymptotic approximation of ln(x) in the complex plane, maybe showing where the zeros are, and how to derive the a0 coefficient? I generated the Taylor series and the zeros and the complex plane graph for the fake function, which is visually the same as before. The approximation is the zeros of f(x) are approximately the zeros of \( \exp(x)\ln(x)+x^{-1}-x^{-2} \)

zeros of the asymptotic Taylor series of f(x), where \( \ln(x)\approx \exp(-x)f(x)\; \) these zeros are exactly 1 greater than the zeros of the \( \exp(x)\ln(x+1) \) asymptotic I posted on mathstack, in answering Mick's question.

Code:

zero[1]= 0.6763550778654
zero[2]= -3.014193883364 + 6.790134224598*I
zero[3]= -3.782847137525 + 13.31987455364*I
zero[4]= -4.248507475714 + 19.72207503441*I
zero[5]= -4.583409659397 + 26.07792021174*I
zero[6]= -4.844835249373 + 32.41104823228*I
zero[7]= -5.059115679636 + 38.73113851434*I
zero[8]= -5.240572105211 + 45.04296385204*I
zero[9]= -5.397864071944 + 51.34917648054*I
zero[10]= -5.536627830306 + 57.65137956884*I
zero[11]= -5.660737008441 + 63.95060483952*I
zero[12]= -5.772967940649 + 70.24754933585*I
zero[13]= -5.875378096927 + 76.54270265230*I
zero[14]= -5.969534334368 + 82.83641978994*I
zero[15]= -6.056657240617 + 89.12896507913*I
zero[16]= -6.137716050140 + 95.42053981217*I
zero[17]= -6.213493139944 + 101.7113002628*I
zero[18]= -6.284629098684 + 108.0013698029*I

Taylor series for f(x), where \( \ln(x)\approx \exp(-x)f(x)\;\; \) The error term for x=100 is 3.7E-46 An approximation for large n is \( a_n\approx \ln(n+0.5)/n! \)
notice that a0=-EulerPhi....

Code:

{fakelnx ~= exp(x)ln(x)
       -0.57721566490153286060651209008240243104215933593992
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+x^300*  1.8641751918061524199262751313929480833397412117845 E-614; }

Complex plane plot from -40, to +100, real, and -10 to +100 imag, grids every 10 units