1 / 31

Harmonic Functions

Harmonic Functions. MTH 324. Lecture # 13. Previous Lecture’s Review. Essential condition for a function to be analytic Cauchy-Riemann equations in polar coordinates Sufficient condition for analyticity using polar coordinates. Lecture’s Outline.

mchristensen
Download Presentation

Harmonic Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Harmonic Functions MTH 324 Lecture # 13

  2. Previous Lecture’s Review • Essential condition for a function to be analytic • Cauchy-Riemann equations in polar coordinates • Sufficient condition for analyticity using polar • coordinates

  3. Lecture’s Outline • Second order partial derivatives • Laplace equation • Harmonic functions • Harmonic conjugate • construction of analytic function

  4. Second order partial derivatives:

  5. Two dimensional Laplace equation:

  6. Harmonic function: Remark:

  7. Example: solution:

  8. Theorem: Proof.

  9. Example: Solution:

  10. Example: Solution:

  11. Harmonic conjugate functions

  12. Example: Solution:

  13. Example: Solution:

  14. Direct method of constructing an analytic function: Working Rules:

  15. Example: Solution:

  16. Harmonic equation in polar form: Example:

  17. Solution:

  18. References • A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan. • Complex variables and applications by James Brown and Ruel Churchill • Fundamentals of complex Analysis by Muhammad Iqbal

More Related