An imaginary tale by Paul Nahin

I first encountered this book in 2009 at Morris library and I still remember at that time I was trying to read it and dozzed off. Over the years Paul Nahin has become my favorite author. He is a very enthusiastic Mathematical expositor and his prerequisites makes you feel smart that you are not just reading a feel good math book written for everyone. There is a certain hubris that you are among the chosen few who have mastered basic math and do not shy away from seeing an integral or partial derivative.

Chapter 7: The Nineteenth Century, Cauchy and the Beginning of Complex Function Theory

This chapter gives an overview of some preliminary topics in Complex function theory. We get to know about Cauchy Riemann equation, Lagrange formula.

Cauchy Riemann  (CR) Equations

Let be a complex function and if we take its derivative along the axis we get and if we take its derivative along the axis we get  , since both of these derivatives are equal (a derivative exists if we get the same value of slope independent of path). So we have and .

Laplace Equation:

If we differentiate CR equations and add we get what is called as Laplace equation

which is

which is .

Adding we get . Similarly by differentiating other way we get .

After that there is a brief introduction of Cauchy’s first Integral theorem and Cauchy’s 2nd integral theorem are given. They are proved using Green’s theorem and we get to know the calculation of residues and how to use them to find integral.s

 

Appendix

The Fundamental theorem of Algebra

Gauss was not the first person to attempt the proof. People like d’Alembert, Euler and Lagrange had all tried but failed.

Descartes believed in that and had mentioned that though it was not a proof. His line of reasoning was “Every root ” of the polynomial equation must appear as the factor in the factorization of . And if has degree such factors (and hence n such roots) to give the required term.