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
- 7. Conclusion: path dependence implies nowhere differentiable
- 8. We proceed to consider two cases…
- 9. Conclusion: differentiable only at the origin
- 10. Conclusion: differentiable everywhere Sounds ‘entire’ to me…
- 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.
- 17. Let where
- 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
- 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
- #24: George
- #25: 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.
- #26: Emily
- #27: Rob
- #28: Rob
- #29: Nick