Comparison on calculus of real and complex numbers

You will do a
review on similarities and differences between 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 this differences brings to other fields of mathematics and to science.

OBJECTIVE: 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 Objective, 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.

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, Birkhauser Boston, Inc., Boston, ¨
MA, 2006.

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