CAS EXPERIMENT. Graphing Potentials. Graph the potentials in Probs. 5 and 7 and for

Integrate \(|z|^{2}\) around the unit circle. Does your result contradict Theorem 1? Text Transcription: |z|^2

VERIFY THEOREM 1 for the given \(F(-), \tau_{0}\), and circle of radius 1 . \((z+1)^{3}, z_{0}=2\) Text Transcription: F(-), tau_0 (z + 1)^3, z_0 = 2

VERIFY THEOREM 1 for the given \(F(-), \tau_{0}\), and circle of radius 1 . \((z-2)^{2}, z_{0}=\frac{1}{3}\) Text Transcription: F(-), tau_0 (z - 2)^{2}, z_0 = 1 / 3

VERIFY THEOREM 1 for the given \(F(-), \tau_{0}\), and circle of radius 1 . \(10 z^{4}, z_{0}=0\) Text Transcription: F(-), tau_0 10z^4, z_0 = 0

VERIFY THEOREM 2 for the given \(\Phi(x, y)\), \(\left(x_{0}, y_{0}\right)\) and crrcle of radius 1 . (x - 2) (y - 2), (4, -4) Text Transcription: Phi(x, y) (x_0, y_0)

VERIFY THEOREM 2 for the given \(\Phi(x, y)\), \(\left(x_{0}, y_{0}\right)\) and crrcle of radius 1 . \(x^{2}-y^{2},(3,8)\) Text Transcription: Phi(x, y) (x_0, y_0) x^2 - y^2, (3,8)

VERIFY THEOREM 2 for the given \(\Phi(x, y)\), \(\left(x_{0}, y_{0}\right)\) and crrcle of radius 1 . \(x^{3}-3 x y^{2},(1,1)\) Text Transcription: Phi(x, y) (x_0, y_0) x^3 -3xy^2, (1, 1)

Derive Theorem 2 from Poisson's integral formula.

Graphing Potentials. Graph the potentials in Probs. 5 and 7 and for three other functions of your choice as surfaces over a rectangle or a disk in the xy-plane. Find the locations of maxima and minima by inspecting these graphs.

Maximum Modulus of Analytic Functions. (a) Verify Theorem 3 for (i) \(F(z)=z^{2}\) and the square \(4 \leqq x \leqq 6,2 \leqq y \leqq 4\), (ii) \(F(z)=e^{3 z}\) and any bounded domain, (iii) F(z) = sin z and the unit disk. (b) F(x) = cos x ( x real ) has a maximum 1 at 0 . How does it follow that this cannot be a maximum of |F(z)| = |cos z| in a domain containing z = 0 ? (c) \(F(z)=1+|z|^{2}\) is not zero in the disk \(|z| \leqq 4\) and has a minimum at an interior point. Does this contradict Theorem 3? (d) If F(z) is analytic and not constant in the closed unit disk \(D:|z| \leqq 1\) and |F(z)| = c = const on the unit circle, show that F(z) must have a zero in D. Can you extend this to an arbitrary simple closed curve? Text Transcription: F(z) = z^2 4 leqq x leqq 6, 2 leqq y leqq 4 F(z) = e^{3z} F(z) = 1 +|z|^2 |z| leqq 4 D:|z| leqq 1

Find the location and size of the maximum of |F(z)| in the unit disk \(|z| \leqq 1\). \(F(z)=z^{2}-1\) Text Transcription: |z| leqq 1 F(z) = z^2 - 1

Find the location and size of the maximum of |F(z)| in the unit disk \(|z| \leqq 1\). F(z) = az + b(a, b complex ) Text Transcription: |z| leqq 1

Find the location and size of the maximum of |F(z)| in the unit disk \(|z| \leqq 1\). F(z) = cos 2z Text Transcription: |z| leqq 1

Verify the maximum principle for \(\Phi(x, y)=e^{x} \cos y\) and the rectangle \(a \leqq x \leqq b .0 \leqq y \leqq 2 \pi\) Text Transcription: . Phi(x, y) = e^{x} cos y a leqq x leqq b, 0 leqq y leqq 2 pi

Do \(\Phi\) and a harmonic conjugate \(\Psi\) of \(\Phi\) in a region R have their maximum at the same point of R? Text Transcription: Phi Psi

Find the location \(\left(u_{1}, v_{1}\right)\) of the maximum of \(\Phi*=e^{u} \cos v\) in \(R*:|w| \leqq 1, v \geqq 0\), where w = u + iv. Find the region R that is mapped onto R* by \(w=f(z)=z^{2}\). Find the potential in R resulting from \(\Phi^{*}\) and the location \(\left(x_{1}, y_{1}\right)\) of the maximum. Is \(\left(u_{1}, v_{1}\right)\) the image of \(\left(x_{1}, y_{1}\right)\)? If so, is this just by chance? Text Transcription: (u_1, v_1) Phi* = e^u cos v R*:|w| leqq 1, v geqq 0 w = f(z) = z^2 (x_1, y_1) (u_1, v_1) (x_1, y_1)