Exercise. The linear transformation [MATH]z \rightarrow t = (-0.2+0.5i)+1.3z \rightarrow w = \frac{1}{2}(t+\frac{1}{t})[/MATH] applies the circumference [MATH]|z|=1[/MATH] in a Kutta-Joukowski profile J in the w-plane. If [MATH]z(w)[/MATH] is the inverse relation then [MATH]G(w) = F(z(w))[/MATH] will be a airfoil. Where does the derivate anulates[MATH] (=0)[/MATH] in J, for [MATH]k=0[/MATH]?
Previous Information. [MATH]f(z) = w = \frac{1}{2}(z+\frac{1}{z})[/MATH] ; [MATH]F(z) = f(z) + ikln(z), k\geq 0[/MATH]
What I've done so far: By doing some research, I have found out that the linear transformation suggests a circle in the w-plane and I have also inverted the relation getting the result [MATH]z(w) = w - \sqrt{(w^2-1)}[/MATH]. Also, for [MATH]k=0[/MATH] I have that [MATH]F(z) = f(z) \Rightarrow G(w) = F(z(w)) = w \Rightarrow \frac{dF}{dw}=1[/MATH] which is never [MATH]0[/MATH] .This result obviously means that I am doing something wrong, but I can't find out where.
Any help would be apreciatted,
Best regards
Previous Information. [MATH]f(z) = w = \frac{1}{2}(z+\frac{1}{z})[/MATH] ; [MATH]F(z) = f(z) + ikln(z), k\geq 0[/MATH]
What I've done so far: By doing some research, I have found out that the linear transformation suggests a circle in the w-plane and I have also inverted the relation getting the result [MATH]z(w) = w - \sqrt{(w^2-1)}[/MATH]. Also, for [MATH]k=0[/MATH] I have that [MATH]F(z) = f(z) \Rightarrow G(w) = F(z(w)) = w \Rightarrow \frac{dF}{dw}=1[/MATH] which is never [MATH]0[/MATH] .This result obviously means that I am doing something wrong, but I can't find out where.
Any help would be apreciatted,
Best regards