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Advanced Engineering Math II phần 3

Hãy suy nghĩ về nó như là bản đồ ở trên trong đảo ngược: Những hình ảnh trên bên phải cho thấy nửa trên lĩnh vực, và chúng tôi nhận được: (F) f: C Cộng đồng Kinh tế ASEAN; f (z) = sin z II Đối với điều này là hữu ích để nhớ rằng f (x + iy) = sinx coshy + i cosx sinhy và chúng tôi thấy rằng nó thực hiện điều này Æ | Think of it as the above map in reverse The above picture on the right shows the top half the domain and we get F f C-C fz sin z For this it is useful to remember that f x iy sinx coshy i cosx sinhy and we find out that it does this -n 2 n 2 -1 1 the next block over n 2 x n goes underneath the axis and then it repeats as we go across the left-hand G f C - 0 C f z z or w 1 . Look at what happens to the general point z x iy 1 x - iy . w 1 u iv x iy x y A vertical line in the w-plane corresponds to u k x _. 2 2 k a constant x y But this is the equation to a circle For instance taking k 1 gives the circle center 1 0 radius 1. In general all these circles pass through the origin where f is not defined . since the above equation when cross-multiplied is satisfied by 0 0 . Similarly horizontal lines also correspond to circles but this time centered on the y-axis. In general we have the following 21 Proposition 6.4 The transformation w 1 z takes circles or straight lines to circles or straight lines. Proof One can represent circles and straight lines by A x2 y2 Bx Cy D 0 Now x2 y2 zz and x z z 2 y z-z 2i. So the above equation can be rewritten as A-. B z z C z-z Azz 2 2i D 0 Now write this in terms of w 1 z. Substituting z 1 w z 1 w and multiplying by ww gives us A B w w 2 C w-w 2i Dww 0 or A Bu - Cv D u2 v2 0 again the equation of a circle or straight line. Ú More generally Theorem 6.5 Every map of the form f z circles or straight lines az b cz d takes circles or straight lines to Proof We can manipulate f z to write it in the form fz a 1 J L c d z- which is a composite affine maps and inversions. Continuing with the examples. G f z z is conformal everywhere except at the origin. In fact it doubles angles at the origin. Some reverse ones Examples A Find a complex function that maps the upper half plane into the wedge 0 Arg z n 4. B Ditto for the Strip 0 y n Wedge 0 Arg w n 4. Look at the exponential map. .