3) Calculate the residues at each of the singularities of 32-1 (22)(2+1)? (+1)
6. (16 points) Sections 6.1-6.3 For each of the singularities of the following functions: i. find the residues ii. find the principal parts iii. classify the singularities as one of the types: removable, pole or essential sin 22 a. cosh b, C. 2 + 2 2² - 32 d. ze
Thanks (c) Locate and classify all the singularities of the function Then compute the residues at each of these singularities. (d) Give an example of a function with a pole of order 5 at 1 -2i, an essential singularity at 1, and a removable singularity at 0. Justify all assertions.
Exercise 12: Residues and real integrals (a) [6+4 points) Compute the residues for all isolated singularities of the following functions (i) f(2)== (2-) tan(2), (i) 9(2):= z2 sin () (b) (4+6+5 points) Compute (using the Residue theorem) (1) cos(72) ( d, A3 := {z € C:<3), 243 := {Z EC: | = 3}, 34, (2-1)(2 + 2)2(2-4) : 43 € C:21 <3}, po 12 To (x2 + 4)2 da, 24 2 + 4 cosat. J 5 + 4 sin(t)
pls help (6) Show that all singularities of f(2) and evaluate sin() are simple poles. Find the residues 22:-1 ffo where is the circle 2| = 3 in ced.
Find the residues of the following functions at 1. 15. 1/(22 – 1)(2 + 2) 16. (23 – 1)(2 + 2)/(2* – 1)2 at
PLEASE SHOW ALL WURI UADI UDD (2+1)e1- 1. (10 points) Classify the singularities of the function f(z) = 72 (22 - 1)(2+3)2
1 - 22) sin(Tz) Consider fe)() a. Find all isolated singularities of f in C and classify each as removable, a pole (specity the order), or essential. b. Explain, with reference to part (a), why f has a series expansion of the form Σ000 ch3k valid near 0. c. Find co d. What is the radius of convergence of the series in part (b)?
2. Caleulate the residues at the indicated poles coS 2 a)- z=0; il , 2 2 sin z (c) 따가, 1+22)22j. 2. Caleulate the residues at the indicated poles coS 2 a)- z=0; il , 2 2 sin z (c) 따가, 1+22)22j.
Determine all the singularities of the follow- ing functions and classify them. (a) COS(2) (b) A -1 (C) (271) (d) - COS(22) (d) 17 – 1)2(72 +1)
ex4.13 Ex. 4.13. Calculate the residues at the poles indicated 1 COS 2 2 (a) 2 (b) sin 2 (c) 2 0; (1+z2)2