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6.1. (a) Write the function f(z) = 2³ +1 in the form f(z) = u(x, y) +iv(x, y). (b) Write the function f(z) = x + 2iy2 as a function involving z and z. 1 (c) Find f(1 - 2i) if f(2)= + Z. 22-1

6.1. (a) Write the function f(z) = 2³ +1 in the form f(z) = u(x, y) +iv(x, y). (b) Write the function f(z) = x + 2iy2 as a function involving z and z. 1 (c) Find f(1 - 2i) if f(2)= + Z. 22-1

Advanced Engineering Mathematics
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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Exercises 6.1. (a) Write the function f(z) = z³+1 in the form f(z) = u(x, y) +iv(x, y). (b) Write the function f(z) = x + 2iy2 as a function involving z and z. 1 (c) Find f(1-2i) if f(x) = +2. 2² - 1 6.2. Determine where the following functions are analytic and find their derivatives: (a) f(z) = (2+1)4¹ + 1/z. 1 (z + 2i) ² (z² + 1) (b) f(2)=
Transcribed Image Text:Exercises 6.1. (a) Write the function f(z) = z³+1 in the form f(z) = u(x, y) +iv(x, y). (b) Write the function f(z) = x + 2iy2 as a function involving z and z. 1 (c) Find f(1-2i) if f(x) = +2. 2² - 1 6.2. Determine where the following functions are analytic and find their derivatives: (a) f(z) = (2+1)4¹ + 1/z. 1 (z + 2i) ² (z² + 1) (b) f(2)=
68 6 Analytic Functions (c) f(2)= 10(2iz² + 5) 6. (d) f(2)= x³ - 3xy² +1+i(3x²y - y³). (e) f(z) = |2|². 6.3. If fu+iv is non-constant and analytic in an open set U, which one of the following are analytic as well? (a) g = u - iv, (b) h=v+iu, (c) k=-u- iv, (d) l= iu - v. 6.4. Using the Cauchy-Riemann criterion show that f(2)= 2³ is entire. 6.5. Let f(z) = (x³ + 3x²y + y³) +i(-x³-3xy²). (a) Use Cauchy-Riemann equations to find all points at hich tiable. Show these points graphically. (b) Is there any open set in which f is differentiable? If yes, find one such open set. Is f analytic anywhere? 6.6. For each case find the region in which f is analytic and find f'. (a) f(2)= zn/n!. n=0² (b) f(2)=non!z". (-1)n-1 (c) f(z) = n 1 (d) f(2)=n=0 2n (z − 1)n 1 (e) f(2)= x=0 n+1 n+i 2n 6.7. Suppose that the radius of convergence of the power series o anz is equal to 1. If we know that f(k) (0) = 0 for all k ≥ 1, and f(0) = 1, find f(1/2+1). 2n=0 2n 6.8. Show that f(2)= Eno(z+1)/n³ is analytic in D₁(-1) and find f'(2). 6.9. Find where the function f(2)=z- is analytic. Show that f'(z) = 6.10. Show that f(z) = equation 23 25 27 3 5 1 z²+1 22n (n!)² 2² f"(z) + z f'(z) = 42² f(z). differen- 2n=0 is entire and satisfies the differential
Transcribed Image Text:68 6 Analytic Functions (c) f(2)= 10(2iz² + 5) 6. (d) f(2)= x³ - 3xy² +1+i(3x²y - y³). (e) f(z) = |2|². 6.3. If fu+iv is non-constant and analytic in an open set U, which one of the following are analytic as well? (a) g = u - iv, (b) h=v+iu, (c) k=-u- iv, (d) l= iu - v. 6.4. Using the Cauchy-Riemann criterion show that f(2)= 2³ is entire. 6.5. Let f(z) = (x³ + 3x²y + y³) +i(-x³-3xy²). (a) Use Cauchy-Riemann equations to find all points at hich tiable. Show these points graphically. (b) Is there any open set in which f is differentiable? If yes, find one such open set. Is f analytic anywhere? 6.6. For each case find the region in which f is analytic and find f'. (a) f(2)= zn/n!. n=0² (b) f(2)=non!z". (-1)n-1 (c) f(z) = n 1 (d) f(2)=n=0 2n (z − 1)n 1 (e) f(2)= x=0 n+1 n+i 2n 6.7. Suppose that the radius of convergence of the power series o anz is equal to 1. If we know that f(k) (0) = 0 for all k ≥ 1, and f(0) = 1, find f(1/2+1). 2n=0 2n 6.8. Show that f(2)= Eno(z+1)/n³ is analytic in D₁(-1) and find f'(2). 6.9. Find where the function f(2)=z- is analytic. Show that f'(z) = 6.10. Show that f(z) = equation 23 25 27 3 5 1 z²+1 22n (n!)² 2² f"(z) + z f'(z) = 42² f(z). differen- 2n=0 is entire and satisfies the differential
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