Exercises involving elementary functions
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1 017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 1 Exercises involving elementary functions 1 This question was in the class test in 016/7 and was worth 5 marks a) Let z + 7 f 1 z) = z 7z + 10 At what points does this function have poles? [1 mark] Express the function f 1 z) in partial fraction form and state the residue at any pole [9 marks] b) Determine the residue at z = 1 of the following function c) Let f z) = z9 + 1 z 7 1 f 3 z) = z + 1) z + ) 3 [7 marks] Determine the partial fraction representation and give the residue at z = This question was in the class test in 015/6 and was worth 4 marks a) Express the function f 1 z) defined below in partial fraction form and state the residue at any pole 3z f 1 z) = z 1)z + ) [9 marks] b) Determine the residue at of the following rational function f z) = z8 z 4 [7 marks] c) Determine the residue at 1 of the following rational function f 3 z) = zz + 3) z + 1) 3
2 017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 3 This question was in the class test in 013/4 and was worth 18 marks a) Determine the partial fraction representation of the function fz) = and give all the residues of the function z + 3 z + 1)z + ) b) Determine the residue at z = of the following function gz) = z 6 z + 1)z + ) [10 marks] 4 The following was in the class test in 014/015 and was worth 5 marks Express the functions f 1 and f defined below in partial fraction form and state the residue of any pole a) b) f 1 z) = z 1 f z) = Determine the residue at z = of the following function [10 marks] z z + 1) f 3 z) = z 10 z z [7 marks] 5 Let Rz) = pz) z ζ 1 ) r 1 z ζ ) r z ζ n ) rn denote a rational function in which ζ 1,, ζ n are distinct points, where each r k 1 is an integer and where pz) is a polynomial which is non-zero at these n points What can you say about the order of the poles of R z) and R z) and what can you say about the residues of the function R z)?
3 017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 3 6 Let qz) = z ζ 1 ) r 1 z ζ ) r z ζ n ) rn where ζ 1,, ζ n are distinct points What can you say about the multiplicity of the zeros of q z) at the points ζ 1,, ζ n? Using a derivation based on partial fractions show that q z) qz) = r 1 z ζ 1 + r z ζ + + r n z ζ n Observe that the result is consistent with the result in an exericise of the previous exercise sheet which involved a proof by induction) 7 This question was in the class test in 016/7 and was worth 6 marks The definition of sinh z is sinh z = ez e z With x, y R express sinhx + iy) in terms of cosh x, sinh x, cos y and sin y 8 By using the definitions show that sin z = 1 i e iz e iz), cos z = 1 e iz + e iz) sinz 1 + z ) = sin z 1 cos z + cos z 1 sin z, cosz 1 + z ) = cos z 1 cos z sin z 1 sin z Further show that ) z1 + z sin z 1 sin z = cos sin z1 z )
4 017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 4 9 The following was in the class test in 014/015 and was worth 14 marks The complex sine function is defined by sin z = 1 i e iz e iz) Describe in Cartesian form the image of each of the following 3 line segments under the mapping w = fz) = sin z a) Γ 1 = { π/ iy : y 0} b) Γ = [ π/, π/] c) Γ 3 = {π/ iy : y 0} State, as concisely as possible, what the image of Γ 1 Γ Γ 3 is under the mapping f 10 Let z = x + iy with x, y R What is the image of the rectangle {z = x + iy : 1 x 1, 0 y π} when we have the function w = expz) = e z 11 This question was in the class test in 016/7 and was worth 10 marks The complex cosine function is such that for x, y R cosx + iy) = cos x cosh y i sin x sinh y Let b > 0 and let z 1 = ib, z = 0, z 3 = π and z 4 = π + ib Describe as concisely as possible the image of the polygonal path z 1 to z, z to z 3 and z 3 to z 4 under the mapping w = cosz) State, as concisely as possible, the image of the line segment joining π/ and π/+ib under the mapping w = cosz) [ marks] 1 This question was in the class test in 013/4 and was worth 10 marks The complex cosine is given by cosx + iy) = cosx) coshy) i sinx) sinhy) for x, y R Describe the lines in the complex z-plane when cosx + iy) is real and describe the part of the complex plane when cosx + iy) is both real and cosx + iy) > 1
5 017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 5 13 Show that if y R then tan π 4 + iy ) = 1 14 Give the definition of the principal value of z α and show that d dz zα = α z α 1 15 a) What are the real and imaginary parts of the principal value of 1 + i ) i b) Determine all the values of 1 1/π 16 This was in the class test in 013/4 and was worth 1 marks State the definition of the principal valued logarithm function and also state the definition of the principal value of z α for any α C and any z 0 With e denoting the usual mathematical constant exp1) = 7188 determine the principal value of the following a) b) Logie) Reie) i )
6 017:11:0:16:4:09 c M K Warby MA3614 Complex variable methods and applications 6 17 This was in the class test in 016/7 and was worth 13 marks State the definition of the principal valued logarithm function State the definition of the principal value of z α for any α C and any z 0 a) Determine the principal value of 1) i b) Let π < α π For all possible α in this interval give the value of Logie iα ) in cartesian form where Log denotes the principal valued logarithm 18 This was in the class test in 015/6 and was worth 10 marks [5 marks] State the definition of the principal valued logarithm function State the definition of the principal value of z α for any α C and any z 0 If z = α = e iθ, θ π, π] then explain why the magnitude of the principal value of z α is e θ sin θ [6 marks]
Exercises involving elementary functions
017:11:0:16:4:09 c M. K. Warby MA3614 Complex variable methods and applications 1 Exercises involving elementary functions 1. This question was in the class test in 016/7 and was worth 5 marks. a) Let
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