Comparison on Calculus of Real and Complex Numbers

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 and Obj.#2 is already completed. Only Obj.#3 needs to be completed. ***

***I will submit the “Obj.#3 – Report” to my instructor and he will give me some feedback about it. Maybe some corrections needed by you at that point. ***

Obj. #1(THIS OBJECTIVE IS ALREADY DONE) 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, …). Write a report on Obj. #1, about 5 – 6 pages with proper citations, and on a separate page(s) the references (only the ones used in the report), in an MS Word file.

Obj. #2(THIS OBJECTIVE IS ALREADY DONE) 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 explicitly

defined sets under the complex function and under its linear approximation).

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?). Write a report on Obj. #2, about 8 – 9 pages with proper citations, and on a separate

page(s) the references (only the ones used in the report), in an MS Word file.

Obj. #3(THIS IS THE OBJECTIVE YOU NEED TO COMPLETE) The aim of this objective is to compare the complex functions and real functions from differentiation 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 analyticity of a function at a point perspective.

Check from a direct application of an analytic function (analytic function is also a

conformal map). You can use solving Laplace’s equation.

Write a report on Obj. #3, about 6 7 pages with proper citations, and on a separate page(s) the references (only the ones used in the report), in an MS Word file.

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