Chap 9. Conformal Mapping

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Presentation transcript:

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

Ex1.

Ex2. Consider two smooth arcs

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

Ex4.

80. Further Properties We know length

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

Ex2.

81. Harmonic Conjugates Recall

P Q

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

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.

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:

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:

Ex2.

Ex3. arctan arctan

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.

Thm: suppose that a transformation

Pf:

Ex.