I have been studying numerical analysis in depth and I am wondering if there is a subject like complex Numerical analysis in which we study the concept of numerically solving complex variable equations, complex integration etc.?
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$\begingroup$ There is, if you google it: maths.usyd.edu.au/u/olver/teaching/NCA/01.pdf | maths.usyd.edu.au/u/olver/teaching/NCA | google.com/… | people.csail.mit.edu/jaffer/III/ComplexMagnitude | kurims.kyoto-u.ac.jp/~okamoto/paper/Publ_RIMS_DE/41-4-42.pdf $\endgroup$– MirkoSep 23, 2019 at 16:29
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2 Answers
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As far as I am aware of, there is no dedicated field "complex numerical analysis" as you describe it. Most of the time, the algorithms for solving a problem with complex variables are essentially the same when dealing with complex variables as when dealing with real variables, or complex problems are easily reduced to real valued problems. For example, Gaussian elimination works exactly the same when dealing with complex variables as when dealing with real values. Integration of a complex-valued function along a path in the complex plane is easily reduced to integrating the real and the imaginary part separately.
There are, however, cases where you can make use of some knowledge from complex analysis. For example, when evaluating the complex exponential, you can use the formula
$$ e^{a + \mathrm{i}b} = e^{a} \cdot e^{\mathrm{i} b } = e^{a} (\cos(b) + \mathrm{i} \sin(b)) $$
which just involves the evaluation of $\exp$, $\sin$, and $\cos$ for real numbers. You could, however, also construct an an approximation to the exponential by exploiting the fact, that $\exp$ is a holomorphic function.
To sum up, I do not think that there is a dedicated field "complex numerical analysis". However, people in numerical analysis make use of results from complex analysis whenever it appears to be useful and they do solve problems with complex variables.
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Depending on what you mean by "concept of numerically solving complex variable equations, complex integration etc." things that come to mind are :
- Homotopy continuation methods
- Complex step differentiation : https://blogs.mathworks.com/cleve/2013/10/14/complex-step-differentiation/
- Numerical conformal mapping : https://archive.siam.org/pdf/news/1297.pdf
You may also be interested in Problem 5 from the SIAM 100 Digit challenge : http://www-m3.ma.tum.de/m3old/bornemann/challengebook/Chapter5/
