6720-2022-05

Let \(C\) be the unit circle centered at the origin. Cauchy’s integral formula implies that \[ f(z)=\frac {1}{2\pi i}\oint _C \frac {f(\xi )}{\xi -z}\,d\xi ,\qquad |z|<1. \]

1. Perform the change of variables \(\xi =e^{i\theta }\) to show that \[ f(z)=\frac {1}{2\pi }\int _0^{2\pi }\frac {f(\xi )\xi }{\xi -z}\,d\theta , \] where \(z\) lies inside the unit circle. Explain why \[ 0=\frac {1}{2\pi }\int _0^{2\pi }\frac {f(\xi )\xi }{\xi -1/\bar {z}}\,d\theta . \]
2. Using the fact that \(\xi =1/\bar {\xi }\) on the unit circle, show that \[ f(z)=\frac {1}{2\pi }\int _0^{2\pi } f(\xi ) \left (\frac {\xi }{\xi -z}\pm \frac {\bar {z}}{\bar {\xi }-\bar {z}}\right )\,d\theta . \]
3. Taking the plus sign in part (b), deduce the Poisson formula for the real part \(u(r,\phi )\) of \(f(z)\) with \(z=re^{i\phi }\): \[ u(r,\phi )=\frac {1}{2\pi }\int _0^{2\pi } \frac {1-r^2}{1-2r\cos (\phi -\theta )+r^2}u(1,\theta )\,d\theta . \] This shows that the real part of \(f(z)\) on the unit circle determines the real part of \(f(z)\) everywhere inside the unit circle.

Use a sector contour with radius \(R\) centered at the origin with angle \(0\le \theta \le 2\pi /5\) to find, for \(a>0\), \[ \int _0^\infty \frac {dx}{x^5+a^5}=\frac {\pi }{5a^4\sin (\pi /5)}. \]

1. Use principal value integrals to show that \[ \int _0^\infty \frac {\cos kx-\cos mx}{x^2}\,dx=-\frac {\pi }{2}(k-m),\qquad k,m>0. \]
2. Deduce from (a) that \[ \int _0^\infty \frac {\sin ^2 x}{x^2}\,dx=\frac {\pi }{2}. \]

Determine the conformal map \(f\) that takes the region \(D\) exterior to two circular cylinders of radius \(R\) to the region outside a circular disc of unit radius centered at the origin (see figure 1). Use the following steps:

1. Show how the map \(\omega =i-2R/z\) takes the region \(D\) to the infinite strip \(0<\operatorname {Im}\omega <2\).
2. Show how the map \(\xi =e^{\pi \omega /2}\) takes the infinite strip to the upper-half complex \(\xi \)-plane.
3. Show how the map \(\eta =(\xi +i)/(\xi -i)\) takes the upper-half complex \(\xi \)-plane to the region outside the unit disc, that is, \(|\eta |>1\).
4. Find \(f\).

Figure 1 description: The map \(f\) sends the \(z\)-plane region exterior to two vertically tangent circular cylinders of radius \(R\) to the \(\eta \)-plane region exterior to the unit disc. The original figure shows the two tangent circles on the left, an arrow labeled \(f\), and a unit circle in the \(\eta \)-plane on the right.

Figure 1: Find the conformal map \(f\)

1. Discuss the flow pattern around a circular obstacle associated with the complex potential \[ \Omega (z)=u_0\left (z+\frac {a^2}{z}\right )+\frac {i\gamma }{2\pi }\log z. \]
2. Show that \(r=a\) is a streamline.
3. Determine the asymptotic velocity when \(z\to \infty \). Sketch the flow when \(\gamma =0\).
4. Use the Blasius formula \[ F_x-iF_y=\frac {1}{2}i\rho \int _C \left (\frac {d\Omega }{dz}\right )^2\,dz \] to determine the lift on the obstacle when \(\gamma \neq 0\).