Problem 1537
At what complex numbers is \(\displaystyle \im z\cdot\re^2 z\cdot i+\overline{z}\) differentiable? Difficulty: 3. |
Problem 1538
At what complex numbers is Difficulty: 3. |
Problem 1539
At what complex numbers is \(\displaystyle |z|^2-(2+i)\bar{z}\) differentiable? Difficulty: 3. |
Problem 1540
Do these functions satisfy the Cauchy-Riemann equations? \(\displaystyle (x^2+y^2,2xy); \qquad (x^2-y^2,2xy); \qquad (e^x\cos y, e^x\sin y). \) Difficulty: 3. |
Problem 1541
Show that \(\displaystyle f(x, y)= \sqrt{|xy|}\) is not differentiable at \(\displaystyle 0\) eventhough it satisfies the Cauchy-Riemann equations there. Difficulty: 3. |
Problem 1542
Let \(\displaystyle f\) be regular on the domain \(\displaystyle D\) with image \(\displaystyle D'\). Assume that \(\displaystyle f\) is injective and let the area of \(\displaystyle D'\) be \(\displaystyle A(D')\). (a) Prove that \(\displaystyle A(D')=\displaystyle\int\limits_D |f'(z)|^2\dx\dy. \) (b) Compare with the theorem on \(\displaystyle \RR^2\to\RR^2\) functions. Difficulty: 5. |
Problem 1536
Apply the conformal property of complex differentiable functions to the Zhukowsky map to show that the ellipses and hyperbolas with foci \(\displaystyle -1\) and \(\displaystyle 1\) intersect each other orthogonally. Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |