📂Complex Anaylsis

Trigonometric Functions as Conformal Mappings

Theorem ¹

The conformal mapping $w = f(z) = \sin z$ maps the vertical line $y=k$ to an ellipse and the horizontal line $x = k$ to a hyperbola.

Proof

Letting $$ z = x + iy \\ w = u + i v $$ we have $$ u = \sin x \cosh y \\ v = \cos x \sinh y $$ Letting $y = k$, $$ {{ u^2 } \over { \cosh^{2} k}} = \sin^{2} x \\ \displaystyle {{ v^2 } \over { \sinh^{2} k}} = \cos^{2} x $$ Adding both sides gives $$ {{ u^2 } \over { \cosh^{2} k}} + {{ v^2 } \over { \sinh^{2} k}} = 1 $$ which is the equation of an ellipse. Letting $x = k$, $$ {{ u^2 } \over { \sin^{2} k}} = \cosh^{2} y \\ \displaystyle {{ v^2 } \over { \cos^{2} k}} = \sinh^{2} y $$ Subtracting one side from the other gives $$ {{ u^2 } \over { \sin^{2} k}} - {{ v^2 } \over { \cos^{2} k}} = 1 $$ which is the equation of a hyperbola.

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