Presentation delivered by students at the University of Leicester on complex differentiablility and analyticity as part of the Complex Analysis module (third year).
Complex Analysis - Differentiability and Analyticity (Team 2) - University of Leicesterr
1. Department of
Mathematics
Year
2013
Lecturer: Dr. Dimitrina Stavrova
Alex Bell | Emily Thorne | George Mileham | Hugh Daman | Joel Duncan
Laura Mulligan | Manij Basnet | Robert Paul Sanders | Shamini Rajan | William Yong
2. • Complex Derivative
• Cauchy-Riemann Equations
• Analyticity
Alex
Will
Manij
Shamini
Joel
Laura
Hugh
Robert
Emily
George
16. The Cauchy-Riemann Relations are:
These give necessary conditions for the existence of a complex derivative. We also
need the first order partial derivatives to be continuous to ensure differentiability.
18. Therefore, when we equate these from both directions, the following must hold
19. Given that
are satisfied
and
find where the Cauchy-Riemann relations
is satisfied nowhere
Conclusion: Cauchy-Riemann equations
are satisfied nowhere
20.
21.
22. Given that
relations are satisfied
and
find where the Cauchy-Riemann
Conclusion: Cauchy-Riemann equations
are satisfied on the whole of
Editor's Notes
George
EmilyGive definitionAll polynomial functions of a complex variable are entire. The proof for this uses the Sum rule on the power series notation of the polynomial, and is example number 4.6 in our notes.The complex sinusoidal function, shown here with its alternative exponential form, is infinitely differentiable everywhere, and consequentially an entire function. A vector plot of sine (z) is shown in the graphic on the right.