📂Complex Anaylsis

Conformal Mapping by Trigonometric Functions

Theorem ¹

Conformal mapping $w = f(z) = \sin z$ maps vertical lines $y=k$ to ellipses and horizontal lines $x = k$ to hyperbolas.

Proof

Suppose $$z = x + iy \\ w = u + i v$$, then $$u = \sin x \cosh y \\ v = \cos x \sinh y$$. Let $y = k$, then $${{ u^2 } \over { \cosh^{2} k}} = \sin^{2} x \\ \displaystyle {{ v^2 } \over { \sinh^{2} k}} = \cos^{2} x$$. Adding both sides, $${{ u^2 } \over { \cosh^{2} k}} + {{ v^2 } \over { \sinh^{2} k}} = 1$$, which becomes the equation of an ellipse. Let $x = k$, then $${{ u^2 } \over { \sin^{2} k}} = \cosh^{2} y \\ \displaystyle {{ v^2 } \over { \cos^{2} k}} = \sinh^{2} y$$. Subtracting both sides, $${{ u^2 } \over { \sin^{2} k}} - {{ v^2 } \over { \cos^{2} k}} = 1$$, which becomes the equation of a hyperbola.

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