(a) Let R be a simply connected region and let f: R→ C be a holomorphic function.Fix a point z in R and define F : R → C viaF(z) =f(s) dc,where C(z) is any contour in R that starts at z* and ends at z.F(zo + h) − F (20) = √2+ f(s) ds,[zo,zo+h]ii. Show that for every ɛ > 0 there is some 8 > 0 such that if h = C satisfies0 < |h| < 8, we have√2-20+h) -(ƒ($) — ƒ(zo)) dz| <ɛ|h|

Answered: Image /qna-images/answer/69acbe2f-3ddf-431d-832a-86b16d0b8b76.jpg

Answered: (a) Let R be a simply connected region…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2x/5+5x=22/15
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4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
Determine a so that u(x,y) = e-πx cos ay is harmonic and finda) the harmonic conjugate function v(x,y) of u(x,y). b) the corresponding analytic function f(z).(f = u+iv)
c) Let the set A = {-1,0, 1} and consider the following function: f3 A × A → A × A given by f(x,y) = (y7,x). Decide whether this function is injective, surjective, a bijection. f3 is an injection: Not Answered • ƒ3 is a surjection: Not Answered f3 is a bijection: Not Answered d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following function: f₁ : P(A) → P(A) given by ƒ(B) = {1, 2, 3}\B. Decide whether this function is injective, surjective, a bijection. f4 is an injection: Not Answered f4 is a surjection: Not Answered • f is a bijection: Not Answered
) Let f(z) = (z − ₁)(z − 3₂)... (zn), where ₁, 2,...,n E C. If y is a positively oriented simple close contour closing every ; in its interior, prove that $ dz ƒ(z) = 0.
Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f:d]. 1 2 - 1 3 1 2 2 - 1 1 1 1 O A. f(x1,X2.X3,X4) = 2x2 - X3 + 5×4, d=5 B. (x1,X2,X3,X4) = 5x, + 6x2 + 2×3 + 5x4, d= 10 O C. f(X1,X2,X3,X4) = x1 - 2×2 – 6×3 + 13×4, d= 12 OD. O D. f(x1,X2.X3.X4) = - 5x, + 2x2 – 6x3 + 5×4, d=0
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Consider the work-form mapping w F₁dx+Fżdy+F;dz where F = (F₁, F2, F3). Given f, a, b, c are all functions of t, x, y, z, calculate the exterior derivative of a = fdt + wÃ where A = (a, b, c). =
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