The question is about Calculus of Residues. (Integration of functions using complex functions).z is a complex number.One of the photos shows the solution. Use only the given solution.Solve parts d and f. 21.4 Functions of Trigonometric FunctionsThe third typc of integral wc can cvaluate using the residuc theorcm involvcsonly trigonometric functions and is typically of the formF(sin 0, cos 0) d0,where F is some (typically rational) function³ of its arguments. Since 0 variesfrom 0 to 27, we can consider it as the angle of a point z on the unit circle3 Recall that a rational function is, by definition, the ratio of two polynomials.21.4 Functions of Trigonometric Functions535centered at the origin. Then z =eit and e-io = 1/z, and we can substitutecos 8 = (z + 1/z)/2, sin 0 = (z – 1/2)/(2i), and dO = dz/(iz) in the originalintegral to obtain1/z z +1/zizdzF2iCThis integral can often be evaluated using the method of residues. 21.5. Evaluate each of the following integrals by turning them into contourintegrals around the unit circle.21.5 Problems53727T2Tdodo(a)(b)where a > 1.5 + 4 sin 0a + cos 027Tdodo(c)(d)where a, b > 0.1+ sin? 0(a + b cos? 0)²

Given : (d) ∫02πdθa+bcos2θ2 where a, b > 0. To Find: Integral by Cauchy residue.

Answered: (b) 2π 62T ᏧᎾ 0 a + cos 0 C2T 10 where…

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(b) 2π 62T ᏧᎾ 0 a + cos 0 C2T 10 where a > 1.

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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Question

The question is about Calculus of Residues. (Integration of functions using complex functions).

z is a complex number.

One of the photos shows the solution. Use only the given solution.

Solve parts d and f.

21.4 Functions of Trigonometric Functions
The third typc of integral wc can cvaluate using the residuc theorcm involvcs
only trigonometric functions and is typically of the form
F(sin 0, cos 0) d0,
where F is some (typically rational) function³ of its arguments. Since 0 varies
from 0 to 27, we can consider it as the angle of a point z on the unit circle
3 Recall that a rational function is, by definition, the ratio of two polynomials.
21.4 Functions of Trigonometric Functions
535
centered at the origin. Then z =
eit and e-io = 1/z, and we can substitute
cos 8 = (z + 1/z)/2, sin 0 = (z – 1/2)/(2i), and dO = dz/(iz) in the original
integral to obtain
1/z z +1/z
iz
dz
F
2i
C
This integral can often be evaluated using the method of residues.

21.5. Evaluate each of the following integrals by turning them into contour
integrals around the unit circle.
21.5 Problems
537
27T
2T
do
do
(a)
(b)
where a > 1.
5 + 4 sin 0
a + cos 0
27T
do
do
(c)
(d)
where a, b > 0.
1+ sin? 0
(a + b cos? 0)²

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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