12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to...
6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that Re /(z)- Re g(z) for all z e B. Show that J-g + ία in D, where α is a real constant. 6. Let D be a bounded domain with boundary B. Suppose that f and g are both analytic on D and continuous on Du B, and suppose further that...
11. (8)(a) Suppose that f and g are analytic branches of zt on a domain D such that 0 g D Show that there is a fifth root, wo, of 1 such that f(z)-wog(2) for all E D. I suggest considering h(z) f (z)/g(z) (b) Now suppose that D D C(-,0]. Let f be an analytic branch of zt in D such that f (1) 1. Show that f(z) expLog(2)) for all z ED. 11. (8)(a) Suppose that f and...
4[10 pts]. Let f(z) = u (r,0) + iv(r,0) be analytic in a domain D c C which does not contain the origin. Then do the following ones: (a) Show that rurr(r, θ) + rur(r, θ) + u69(r, θ) 0 for all re® E D. (b) Show that (a) is equivalent to the condition that u is harmonic in D (c) Show that the function (in|e )2-[Arg( a(z) z)]2,-π < Arg(z) < π, 4[10 pts]. Let f(z) = u (r,0)...
4. Suppose (x) (3+4x)+e*. a) Use analytic methods to show f (x is one to one. b) Find () (28) Suppose g (x)--fx-12 + 3e 30-1 5x + e 2-x-8x + 17 . ₩5. c) Use analytic methods to show g(x) is one to one. d) Find (g") (4) 3x-2 6. Find the equation of the tangent line to the curve y -at the po int (Q,e)
u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented 9 (10) Suppose that f is analytic in the deleted disk B2(0) C be the positi that If(2)l S M<oo for all z e B2(0). If 0 TS circle |zl r. Show that S 1, then let Cr r | 1= f(z) dz = 0. (Hint: why is the value of (1) the same if C, is replaced by C? u(20) for all z...
v e, v, z)dzdydz where f(e.v.)3 Evaluate the triple integral D and Triple Integral Region R Remember that: H(u, t, u)|J(u, v, w)ldududu F(z, y, z)dV Preview t lower limit Preview น upper limit- U lower limit Preview upper limit w lower limit upper limit H(u, o, w)- Preview Preview Ila Preview H(u, e, w)J(u,v, wdudedu Hint: The focus of this problem is on evaluating the integral and using the Jacobian. v e, v, z)dzdydz where f(e.v.)3 Evaluate the triple...
13. Integrate: a. j«x+278)dx 0 b. (dx х c. dx 9+ x d . xdx? +2 dx 2x+1 хр '(x’+x+3) f. I sin (2x) dx g. cos (3x) dx h. ſ(cos(2x)+ + secº (x))dx i. [V2x+1 dx j. S x(x² + 1) dx k. | xe m. [sec? (10x) dx 16 n. .si dx 1+x 0. 16x 1 + x dx 5 P. STA dx 9. [sec xV1 + tan x dx 14. Given f(x)=5e* - 4 and f(0) =...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...