1. Chap 9. Conformal Mapping
不會在某一點 t 停頓
79. Preservation of Angles

5. Ex1.

6. Ex2. Consider two smooth arcs

7. Isogonal mapping : a mapping that preserves the magnitude of the angle bet not necessarily the sense.
Ex3.

8. Ex4.

9. 80. Further Properties
We know
length

10. Large region will be different in shape after transformation.
Ex1.

13. Ex2.

14. 81. Harmonic Conjugates
Recall

16. P Q

17. Ex. Q:
給我一個u,找出對應的v, (必需在simply connected domain)

18. 82. Transformations of Harmonic Functions
Since A function that is harmonic in a simply connected domain always has a harmonic conjugate (sec.81), solutions of (boundary value) problems in such domains are the real or imaginary ports of analytic functions.

19. Ex.
If we can identify a function as the real or imaginary part of an analytic function, then we know it is a harmonic function, But how ?
not easy
Other aid:

20. Thm. Suppose that an analytic function
藉由f and h的條件,
目的:show H(x,y) is
a harmonic function
Simple pf:

21. Ex2.

22. Ex3.
arctan
arctan

23. 83. Transformations of Boundary Conditions
Boundary condition: a function or its normal derivative
(boundary value problem) have prescribed values along the bandore
of a domain.
we can transform a given boundary valne problem in the xy plane into a simpler one in the u v plane and then write the solution of the original problem in terms of the solution obtained from the simpler one.

24. Thm: suppose that a transformation

25. Pf:

29. Ex.