For the mapping of horizontal lines ($z=x+iy_0$ for fixed $y_0$) and vertical lines ($z=x_0+iy$ for fixed $x_0$) under $w(z) = \sin(z)$, are there any general formulas?
What I mean is, is there a general formula like "$z=x_0+iy$ is mapped to a hyperbola with the vertex (something in terms of $x_0$) and the distant from the origin to the vertex (something in terms of $x_0$)"?
I used $$\sin(x_0+iy) = \sin(x_0) \cosh(y) +i \cos(x_0) \sinh(y)$$ and $$ \sin(x+iy_0) = \sin(x) \cosh(y_0) + i \cos(x) \sinh(y_0)$$, but I don't know how to go beyond that to obtain a general formula for the mapping under $w=sin(z)$.
Thank you in advance.
Best Answer
Let $w = u + iv = \sin(x)\cosh(y) + i\cos(x)\sinh(y)$
For $y = y_0$ we have the parametrization
\begin{align} u(x) &= a\sin(x) \\ v(x) &= b\cos(x) \end{align}
where $a=\cosh(y_0)$ and $b=\sinh(y_0)$. This represents a horizontal ellipse with major and minor axes $a$ and $b$, respectively, since $\frac{u^2}{a^2}+\frac{v^2}{b^2}=1$. All ellipses have the focus $\sqrt{a^2-b^2}=1$
Similarly, for $x=x_0$ we have the paremetrization
\begin{align} u(y) &= a\cosh(y) \\ v(y) &= b\sinh(y) \end{align}
where $a = \sin(x_0)$ and $b = \cos(x_0)$. This represents a hyperbola with major axis $a$, since $\frac{u^2}{a^2} - \frac{v^2}{b^2}=1$. The focus of every hyperbola is also $\sqrt{a^2+b^2}=1$