1. Sec 6.17: Conformal Mapping Techniques Solve Our goal in this section

2. Sec 6.17: Conformal Mapping Techniques Complex function 1 f(z) onto 2 f(z) one-to-one

3. Sec 6.17: Conformal Mapping Techniques Example:

4. Sec 6.17: Conformal Mapping Techniques Is conformal if it preserve Def: Conformal mapping 1 angles 2 Orentation

5. Sec 6.17: Conformal Mapping Techniques Let Theorem 6.9: analytic function (D onto K) Conformal mapping Let Theorem 6.10(Riemann Mapping Theorem) : Unit circle

6. Sec 6.17: Conformal Mapping Techniques Remark: A conformal mapping of a domain D onto a domain K will map the boundary of D to the boundary of K.

7. Sec 6.17: Conformal Mapping Techniques Is a function of the form Linear fractional Transformation (bilinear) a, b, c, d complex number and T is a conformal mapping defined on C – {-d/c} Remark:

8. Sec 6.17: Conformal Mapping Techniques Theorem 6.11: T maps a circle to a circle or straight line circle T maps a straight line to a circle or straight line circle

9. Sec 6.17: Conformal Mapping Techniques Theorem 6.12: Find T so that:

10. Example Sec 6.17: Conformal Mapping Techniques Theorem 6.12: How to find T: Think of w =T(z) and solve for w in the equation Find a bilinear mapping that sends the given points to the images indicated.

11. Sec 6.17: Conformal Mapping Techniques How to find T: Theorem 6.10(Riemann Mapping Theorem) : Unit circle Map the right half-plane Re(z)>0 conformally onto the unit disk |w| <1 Select 3 points in z-plane Select 3 images in w-plane We will map boundary to boundary Maintain orentation (counterclockwise) As you walk on the boundary the domain is on your left.

12. Sec 6.17: Conformal Mapping Techniques Given u An integral Solution of the Dirichlet Problem for a Disk Let u be a harmonic function in the unit Disk |z|<1+e slightly larger than the unit the values of u are prescribed on the boundary circle C The complex function f(z) can be written as (see p322 for derivation) Remark

13. Sec 6.17: Conformal Mapping Techniques Solution of Dirichlet Problem by conformal mapping Solution is the real part of The solution of this problem is The real part of the function f(z)

14. Example Sec 6.17: Conformal Mapping Techniques Solution of Dirichlet Problem by conformal mapping The solution of this problemThe real part of the function f(z) Solve the Dirichlet problem for the right half-plane

15. Example Sec 6.17: Conformal Mapping Techniques Solution of Dirichlet Problem by conformal mapping Solve the Dirichlet problem for the right half-plane real Real part