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.__