u(20) for all z e D. Prove tha E C:0<zl<2) and Cr be the positively oriented...
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...
(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0 (14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0
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)...
1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0...
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2
definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0 definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...
U Question 15 "C 7 pts "С If S is the surface of the cylinder E= {(x,y,z) : 32 + y < 4,1523}, oriented outwards, which of the following (after applying the Divergence Theorem) will compute zyz) - dS? 40 O (1 + y2 cos & sin 6)r dr de dz REC O 1988 6%" /*(1 + == sin ®)r dr do dz %%% %%% %%% (r cos 0 + 32 + y2 z cos ( sin 0), dr do...
(1 point) Consider the wave equation 1(1)utt = uzz for-oo < z < oo, t>0 with initial conditions ut (z,0-0 and u(z,0) = /(z), where (2) f(z) = 1 for 0 < z < 1, (3) f(z) =-1 for-1 < z < 0, and (4) f(z) = 0 for all other. The slanting lines in the figure below show the characteristics for this PDE that originate on the z-axis at the points of discontinuity of the initial data f f(x)...
4. Let u be the solution of the Burgers quaslilinear 1st order PDE u, + uu,-0, a(0,x) = g(x) E C2(R2) and suppose that lIgl(R) oo and that u E C"(-T,T) × R). Prove that 2. if g has compact support, then so does u(t,.) for allt E (-T,T); 3. if g 20and has compact support, then 4. if g 2 0 and has compact support, then f(u(t,x))dx= | f(g(x))dx for all f e C([0, oo)).