Let C be the boundary of the square with vertices 0, 1, 1+ i, andi with counterclockwise orientation.(1) Find a parameterisation for C.(2) Evaluate the integral [fa(z) dz,constant.[fa(z) dz, where fa(z)=x² - y² -- ax and a is a complex(3) Find all values of a such that [ fa(z) dz = 0.(4) Let ao be one of the values found in (3). Can we conclude that fa is analytic insideof C? Explain your answer carefully.

Answered: Image /qna-images/answer/64dd6f53-6a54-47a5-b50f-5940f35cdbc7.jpg

Answered: Let C be the boundary of the square…

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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Let A be an n x n matrix over C. Let f be an entire function, i.e. an analytic function on the whole complex plane, for example exp(z), sin(z), cos(z). Using the Cayley-Hamilton theorem we can write ƒ(A) = an−1A”−¹ + an−2A¹−²+...+ a₂Ã² + a₁A+aoIn (1) where the complex numbers ao, a₁, ..., an-1 are determined as follows: Let r(λ) := An−1\”−¹ + an-21²-2+...+ a₂λ² + a₁λ + ao which is the right-hand side of (1) with A¹ replaced by X³, where j = 0, 1, ….., n— 1 of each distinct eigenvalue λ; of the matrix A, we consider the equation ƒ(λj) =r(λj). (2) If X, is an eigenvalue of multiplicity k, for k> 1, then we consider also the following equations ƒ'(A)|a=a; = r′(A)|a=a; ƒ"(A)|a=a; = r″(A)|a=a; ...... f(k-1) (X) = (k-1)(A) (A)|x 1x=x₂ Apply this technique to find exp(A) with с A = = (ε c), C C * |x=x₂ CER, C# 0.
Let T = L(V) where V is a complex inner product space with the standard inner product. Define T₁ = (T+T*), T₂: = 1 :(T – T*) T(v) = 2i and T₁ and T₂ are self-adjoint. (a) Show that T₁T₂ = T₂T₁, then T is normal. (b) Compute T₁ and T₂ when T = L(C²) is given by 0 -J Find eigenvalues and explain if T is diagonalizable.
Define to mapping pi : R2(arrow)R by pi((x,y)) = x. Find the kernel of pi.
Consider a function f(z) that is analytic in the complex plane except for a finite number of isolated singularities. Prove that the integral of f(z) along a closed contour is equal to 2πi times the sum of the residues of f(z) at its singularities within the contour.
6. Find a basis of the null space of each of the differential operators below, and verify that the dimension of the null space is equal to the degree of the operator. (Use complex numbers where necessary for simplicity.) (a) D² − 4D + 3 (b) D² + 2D + 2 (c) D³ (d) Dn (e) Dª + 4.
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Let i =, =, and i =. Please do all of the following. (a) Compute i x w. (b) Compute the scalar triple product (ü x w) - Ë.
43. Linear combination Let u = (1, 2, 1), v = (1,-1,-1), w = (1, 1,-1), and z = (2, -3,-4). Find scalars a, b, and c such that z = au + bv + cw.
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
Let f and g be analytic functions in a domain D. Iff '(z) =g'(z)for all z in D,then show thatf (z) = g(z) + c, where c is a complex constant.
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1 Let f(2) 2- 9 Part (a) Integrate the function f(2)along the curve C in the complex plane given by z = from x = 0.5 to x = 2 Part (b) Integrate the same function along the unit circle in the complex plane, i.e. on Iz| = 1.

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