say we have e^(jx), where j=sqrt(-1) and x can be any real variable. We know that even 2nd order Taylor approximation is not good enough to approximate e^(jx) with arbitrary realization of x. Are there any better approximation methods? Some say Polytopic representation might help, but I am not sure how it can. Thank you.
You are taking about complex exponent which is one of simple complex functions. In is common to use Laurent series for complex functions instead of Taylor series. They use both negative and positive powers. You can read an introduction to them here: http://mathworld.wolfram.com/LaurentSeries.html , https://en.wikipedia.org/wiki/Laurent_series .
But if you work just with real argument x, then
e^(jx)=cos (x) +j Sin(x),
where series for both sin and cos are with alternate signs and converge quickly.
Following the definition, we have e^(ix) = 1 + ix/1! + (ix)^2/2! + (ix)^3/3! + ,,, (i, e. e^(ix) is definited by one Taylor serie everywhere and e^z, z in C, is an analitical function.
The Taylor serie for e^(ix) is not an approximation, it is the definition for it. Your sincerely, Anna Tomova.
Hi dear, Which approximation do you mean? Because we have a few series (Taylor, Laurent and etc) for making an alternative function for e^jx. Sincerely, Mezban
Than you Yuri. I am looking into the Laurent series. Hope it can help.
Thank you for point out the errors, Anna. I meant Taylor approximation, not Taylor series expansion. I have changed the title accordingly.
Thank you Peter. There are a lot of questions. I think I need to think about them thoroughly.
Thank you for asking, Mezban.
I want to find an approximation of e^(ix), with x being a real Gaussian random variable with zero mean and variance \sigma^2. So that, I can compute the expectation of E[e^(jx)] with respect to x with tractable approximation of e^(jx). However, second order Taylor approximation does not give a good approximation when \sigma^2 is large, so I wonder if there are other tractable approximation methods which will facilitate computing E[e^(jx)] .
You may also consider this.
Article Evaluation of the matrix exponential function using finite e...
Well, obviously, the fourier series is the best. But it depends on what do you want.
Best,
DG
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