Plotting Complex Functions.

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Sudharsana Iyengar
Sudharsana Iyengar 2024-8-25
评论： Sudharsana Iyengar 2024-8-28
I intend to do a taylor series expansion at t=0. I would like to plot the coefficients of each Taylor term in the complex plane, using the surf function.
I tried Taylor(F) , Series (F) somewhere there is an error. The function is product of two gaussians, with two parameters mentioned above. This Gaussian contains other constants, Some are symbolic, while some have pre determined values.
Can some one help me out.
Thanks.
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David Goodmanson
David Goodmanson 2024-8-28
编辑：David Goodmanson 2024-8-28
Hi Sudharsana, could you comment on what the coefficients you show above are all about?
Sudharsana Iyengar
Sudharsana Iyengar 2024-8-28
It is a product of two gaussians. These gaussians have different parameters.
The function is expanded around x=0.
Each of the coefficient is integrated in the complex plane.
I hope this helps.

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David Goodmanson
David Goodmanson 2024-8-26
编辑：David Goodmanson 2024-8-28
Hello Sudharsana,
Do you mean
f(t) = exp(-ar/(twC))*exp(-ai/(twC))
or
f(t) = exp(-ar/(twC))*exp(-i*ai/(twC))
^
(per Torsten's first comment)? Either way, this function has an essential singularity at t = 0 in the complex t plane. Assuming the lower line above is correct, which makes sense, then the first factor
exp(-(ar/wC)/t)
has the most to do with the behavior of the function at the origin. Assuming (ar/wC) is positive, then if you approach the origin from the right, the function f(t) and all its derivatives are zero at t = 0+. Since the coefficients of a Taylor series about a point are determined by the derivatives of f(t) at that point, this means that f(t) cannot be expanded in a Taylor series about t = 0. If (ar/wC) is negative then as you approach the origin from the from the right, f(t) --> infinity so again there is no Taylor series.
Which doesn't mean that it can't be approximated. For example, using exp(-1/t) for simplicity
t = 0:.001:4;
f = exp(-1./t);
c = polyfit(t,f,6)
f_fit = polyval(c,t);
I'm not advocating for polyfit which is anyway not a great idea for polynomials of large degree, and I am sure there are better fits but this is just one example.

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