. (a) Show that the function u= 4x2 - 12.xy2 is harmonic and v=12.xy-4v2 is a...
Let W(x, y) be a harmonic function, and also let u(x, y) and v(x, y) be a harmonic conjugate pair. Show by hand that the composite function W(u(x, y), v(x, y)) is also harmonic. Let W(x, y) be a harmonic function, and also let u(x, y) and v(x, y) be a harmonic conjugate pair. Show by hand that the composite function W(u(x, y), v(x, y)) is also harmonic.
The path integral of a function f(x, y) along a path e in the xy-plane with respect to a parameter r is given by 2. fex,y)ds= f(x),ye) /x(mF +y(t" dr , where a sr sb. (a) Show that the path integral of f(x, y) along a path c(0) in polar coordinates where r=r(0), α<θ<β, is Sf(r cos 0,rsin e) oN+( de. (b) Use this formula to compute the arc length of the path r 1+cos0, 0<0 27 The path integral...
7. Let f:D + C be a complex variable function, write f(x) = u(x, y) +iv(x,y) where z = x +iy. (a) (9 points) (1) Present an equivalent characterization(with u and v involved) for f being analytic on D. (Just write down the theorem, you don't need to prove it.) (2) Let f(z) = (4.x2 + 5x – 4y2 + 3) +i(8xy + 5y – 1). Show that f is an entrie function. (3) For the same f as above,...
(a) Let u: R2R be a harmonic function. Show that the function v: R2R defined by is also harmonic. (b) Show that the tranformation maps the positive quadrant Q+-[(x,y): x > 0&y to the upper half plane c)Find the Dirichlet Green function for the positive quadrant + (a) Let u: R2R be a harmonic function. Show that the function v: R2R defined by is also harmonic. (b) Show that the tranformation maps the positive quadrant Q+-[(x,y): x > 0&y to...
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
solve number 3 2. Show that u(x, y) = e = cos 12 + y2 is harmonic on the punctured complex plane D=C\{0}, and find a harmonic conjugate v of it. 1 2 3. Reveal, what is f(x) = u(x,y) + iv(I,y) in Problem 2?
1 R 12. Use the transformation T: u = 5x and v= ky to evaluate the integral ſf xºdA where R is the region bounded on the xy-plane by the ellipse 9x’ +4y? = 36. . Let S be the image of Runder T on the uv-plane. Sketch regions R and S. Set up the integral 7as an iterated integral of a function f(u, v) over region S. Use technology to evaluate the integral. Give the exact answer. R S...
Problem 8. Let f(z) = u(x, y) iv(x, y) be an entire function with real and imaginary parts u(x, y) and v(x, y). Assume that the imaginary part is bounded v(x, y) < M for every z = x+ iy. Prove that f is a constant 1
Question Help LetF=v(xy?) and let C be the path in the xy-plane from (-44) to (4.4) that consists of the line segment from (-44) to (0,0) followed by the line segment from (0,0) to (4.4). Evaluate SA Fodrin two ways a) Find parametrizations for the segments triat make up C C-v*(4-440stst C12(0) = (4) Costs С a) Find parametrizations for the segments that make up C and evaluate the integral b) Use 1(xy] =x?y? as a potential function for F
1 R 12. Use the transformation T: u = -x and very to evaluate the integral [jx?dA where R is the region bounded on the xy-plane by the ellipse 9x + 4y = 36. . Let S be the image of Runder T on the uv-plane. Sketch regions and S. Set up the integral 7as an iterated integral of a function f(u, v) over region S. Use technology to evaluate the integral. Give the exact answer. R S Y