Comparison on Calculus of Real and Complex Numbers

The main goal of this project is to observe the main differences of calculus of complex numbers in C over the real numbers and what advantages these differences brings to other fields of mathematics and to science.

Obj.#1 & Obj.#2 & Obj.#3 are already completed. I am going to put the details about first 3 objectives at the end if you would like to know what were about.

✅ You are going to work on Obj.#4.

✅ ATTENTION: I will submit the “Obj.#4 – Report” to my instructor and he will give me some feedback about it. Then I will share his feedback with you. You should be ready to fix and update the paper based on the feedback if needed.

✅ Check with the example report I attached, and please, arrange the references in a similar format.

✅ Please, use MS Word’s Equation processor to write equations in the report.

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HERE IS THE OBJECTIVE #4.

Obj. #4: The aim of this objective is to compare the complex functions and real functions from techniques of integration perspective and from the series representation perspective.

– Check whether the integration by parts still works for complex integrals.

– There is a mean value theorem for definite integrals. Check whether there is a such theorem for the complex counterpart.

– Check from the power series epension perspective.

– ?!!! Write a report on Obj. #4, about 7−8 pages with proper citations, and on a separate page(s) the references (only the ones used in the report), in an MS Word file.

-Check with the example final report I attached, and please, arrange the references in a similar format.

Thank you.

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You can use the following classical books [PS06],[GK06],[Rud87],[CB84],[MH87],[Ahl78], [Lan99], [Nee97].

REFERENCES

[Ahl78] Lars V. Ahlfors, Complex analysis, third ed., McGraw-Hill Book Co., New York, 1978, An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. MR 510197

[CB84] Ruel V. Churchill and James Ward and Brown, Complex variables and applications, fourth ed., McGraw-Hill Book Co., New York, 1984. MR 730937

[GK06] Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third ed., Graduate Studies in Mathematics, vol. 40, American Mathematical Society, Providence, RI, 2006.

[Lan99] Serge Lang, Complex analysis, fourth ed., Graduate Texts in Mathematics, vol. 103, Springer-Verlag, New York, 1999. MR 1659317

[MH87] Jerrold E. Marsden and Michael J. Hoffman, Basic complex analysis, second ed., W. H. Freeman and Company, New York, 1987. MR 913736

[Nee97] Tristan Needham, Visual complex analysis, The Clarendon Press, Oxford University Press, New York, 1997. MR 1446490

[PS06] S. Ponnusamy and Herb Silverman, Complex variables with applications, Birkh¨auser Boston, Inc., Boston, MA, 2006.

[Rud87] Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.

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Obj.#1 & Obj.#2 & Obj.#3 Details

Obj. #1: The aim of this objective is to compare the set of complex numbers and the set of real numbers from algebraic structural and from geometric perspective (such as closedness, nth-roots, ’rational’ powers, infinity as extending the number system, roots of polynomials, …).

Obj. #2: The aim of this objective is to compare complex complex valued functions(with one complex variable) with their real valued (with one or two real variables) functions. – Check with some fundamental functions such as complex exponential functions. – Look into the the real valued with two-real variables functions encountered in the studying complex functions. – Look into graphing a real and complex function. – Compare linear (real and complex) functions, linear approximation of a real and of a complex function and their meanings (use CAS to compare the images of some explicitlydefinedsetsunderthecomplexfunctionandunderitslinearapproximation). – Check with multi-valued functions, look also into their graphs (for example complex power function, complex logarithm function, complex exponential function) – Check also if all the properties of real powers work also for complex powers. – Check with trigonometric functions. – Check with the inversion function, see that it can map lines to circles and it can be reversed. – Check from the limit of a function perspective. Check from the continuous function perspective (for example in the real case we say there should not be any jumps, do we have the same observation in the complex case?).

Obj. #3: The aim of this objective is to compare the complex functions and real functions from differeantiation and analyticity perspective. – Check with the interpretations of a derivative of a function. – Check with complex version of some real functions and see whether their differentiability changes and how. – Check if any complex functions can be differentiated with a shortcut formulas, like in the real case. – Check from the higher order derivatives of a function perspective. – Check from analiticity of a function at a point prespective. – Check from a direct application of an analytic function (analytic function is also a conformal map). You can use solving Laplace’s equation.