西安电子科技大学：《Complex Analysis》课程教学资源（讲义）Chapter 6 Residue Theory

Ch.6:Residue Thoory Ch.6:Residue Theory 6.1 The Residue Theorem Chapter 6:Residue Theory 6.2 Trigonometric Integrals Over (0,2) Li,Yongzhao State Key Laboratory of Integrated Services Networks,Xidian University 6.3 Improper Integrals of Certain Functions Over(-oo,oo) June7,2009 6.4 Improper Integrals Involving Trigonometric Functions Ch.6:Residue Theory Ch.6:Residue Theory 6.1 The Residue Theorem L6.1 The Residue Theorem Introduction The Residue Theorem In the previous chapters,we have seen how the theory of If f(z)is analytic on and inside a simple closed positively contour integration lends great insight into the properties of oriented contour I except a single isolated singularity.zo. analytic functions lying interior to T.f(z)has a Laurent series expansion The goal this chapter is to explore another dividend of this theory,namely,its usefulness in evaluating certain real ()=a(:--o j=-00 integrals converging to some punctured neighborhood of z0 We shall begin by presenting a technique for evaluating contour integrals that is known as residue theory In particular,the above equation is valid for all z on the small positively oriented circle C continuously deformed from I(as Then we will introduce some application of the theory to the shown in Fig.6.1) evaluating the real integrals

Ch.6: Residue Theory Chapter 6: Residue Theory Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University June 7, 2009 Ch.6: Residue Theory Outline 6.1 The Residue Theorem 6.2 Trigonometric Integrals Over (0, 2π) 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) 6.4 Improper Integrals Involving Trigonometric Functions Ch.6: Residue Theory 6.1 The Residue Theorem Introduction In the previous chapters, we have seen how the theory of contour integration lends great insight into the properties of analytic functions The goal this chapter is to explore another dividend of this theory, namely, its usefulness in evaluating certain real integrals We shall begin by presenting a technique for evaluating contour integrals that is known as residue theory Then we will introduce some application of the theory to the evaluating the real integrals Ch.6: Residue Theory 6.1 The Residue Theorem The Residue Theorem If f(z) is analytic on and inside a simple closed positively oriented contour Γ except a single isolated singularity, z0, lying interior to Γ, f(z) has a Laurent series expansion f(z) = ∞ j=−∞ aj (z − z0)j converging to some punctured neighborhood of z0 In particular, the above equation is valid for all z on the small positively oriented circle C continuously deformed from Γ (as shown in Fig. 6.1)

Ch.6:Residue Thoory Ch.6:Residue Theory L6.1 The Residue Theorem L6.1 The Residue Theorem The Residue Theorem(Cont'd) The Residue Theorem(Cont'd) According to the Continuous Deformation Invariance Theorem (page 231),we have Thus the constant a-1 plays an important role in contour integration.Accordingly,we adopt the following terminology f(z)dz=f(z)dz Definition The last integral can be computed by termwise integration of If f has an isolated singularity at the point zo,then the coefficient the series along C.For allj-1 the integral is zero,and for a-1 of (z-z0)-1 in the Laurent expansion for f around zo is j=-1 we obtain the value 2mia-1 called the residue of f at zo and is denoted by Consequently we have Res(f:z0)or Res(zo) f(z)dz =2wia-1 Ch.6:Residue Theory Ch.6:Residue Theory 6.1 The Residue Theorem L6.1 The Residue Theorem How to Compute the Residue How to Compute the Residue(Cont'd) If f has a removable singularity at z0.all the coefficients of the negative powers of(z-z0)in its Laurent expansion are zero,and so,in particular,the residue at zo is zero Example 2 gives us another way to compute the residue when If f has an essential singularity at zo.we have to use its f is a rational polynomial Laurent expansion to find the residue at zo(See Example 1 on Let f(z)=P(z)/Q(z).where the functions P(z)and Q(z) page 308) are both analytic at zo and Q has a simple zero at zo,while If f has a pole of order m at zo.we have the following P(zo)≠0.Then we have theorem to find the residue P(2z0) Theorem Rs(f:20)=Q(2o If f has a pole of order m at zo.then 1 dm-1 (m-1)!dm(0)mf()] Res(f:2o)=lim万

Ch.6: Residue Theory 6.1 The Residue Theorem The Residue Theorem (Cont’d) According to the Continuous Deformation Invariance Theorem (page 231), we have Γ f(z)dz = C f(z)dz The last integral can be computed by termwise integration of the series along C. For all j = −1 the integral is zero, and for j = −1 we obtain the value 2πia−1 Consequently we have Γ f(z)dz = 2πia−1 Ch.6: Residue Theory 6.1 The Residue Theorem The Residue Theorem (Cont’d) Thus the constant a−1 plays an important role in contour integration. Accordingly, we adopt the following terminology Definition If f has an isolated singularity at the point z0, then the coefficient a−1 of (z − z0)−1 in the Laurent expansion for f around z0 is called the residue of f at z0 and is denoted by Res(f; z0) or Res(z0) Ch.6: Residue Theory 6.1 The Residue Theorem How to Compute the Residue If f has a removable singularity at z0, all the coefficients of the negative powers of (z − z0) in its Laurent expansion are zero, and so, in particular, the residue at z0 is zero If f has an essential singularity at z0, we have to use its Laurent expansion to find the residue at z0 (See Example 1 on page 308) If f has a pole of order m at z0, we have the following theorem to find the residue Theorem If f has a pole of order m at z0, then Res(f; z0) = lim z→0 1 (m − 1)! dm−1 dzm−1 [(z − z0)mf(z)] Ch.6: Residue Theory 6.1 The Residue Theorem How to Compute the Residue (Cont’d) Example 2 gives us another way to compute the residue when f is a rational polynomial Let f(z) = P(z)/Q(z), where the functions P(z) and Q(z) are both analytic at z0 and Q has a simple zero at z0, while P(z0) = 0. Then we have Res(f; z0) = P(z0) Q(z0)

Ch.6:Residue Thoory Ch.6:Residue Theory L6.1 The Residue Theorem L6.2 Trigonometric Integrals Over (0,2x) How to Compute the Residue(Cont'd) Trigonometric Integrals Over [0,2] Our goal of this section is to apply the residue theory to When there are a finite number of isolated singularities inside evaluate real integrals of the form the simple closed positively oriented contour T,we have the 2x following theorem U(cos0,sin0)de (1) Theorem 0 IfI is a simple closed positively oriented contour and f is analytic We use z=ei (0<0<2)to parameterize the closed inside and on I except at the points z1,z2,...,zn inside I,then positively oriented contour |z=1.Then a contour integral f(e)ds-2i Re) can be transformed into a real integral j=1 According to Euler's equation,we have cos0=(e8+e0)/2=(z+z1)/2 sin0=(e0-e0)/2i=(z-z-l)/2i Ch.6:Residue Theory Ch.6:Residue Theory 6.2 Trigonometric Integrals Over (0,2x) -62 Trigonometric Integrals Over (0,2x) Trigonometric Integrals Over [0,2](Cont'd) Trigonometric Integrals Over [0,2](Cont'd) Also taking dz iei do izd0 into account,Eq.(1)can be transformed into a complex contour integration as Of course,the function F must be a rational function of z 2分 Hence,it has only removable singularities(which can be U(cos0,sin0)do=p F(z)dz ignored in evaluation integrals)or poles =1 Consequently,by the residue theorem,our trigonometric where the new integrand F is integral equals 2i time the sum of the residues at those poles @=v(+)(-》是 of F that lie inside the unite circle

Ch.6: Residue Theory 6.1 The Residue Theorem How to Compute the Residue (Cont’d) When there are a finite number of isolated singularities inside the simple closed positively oriented contour Γ, we have the following theorem Theorem If Γ is a simple closed positively oriented contour and f is analytic inside and on Γ except at the points z1, z2, ··· , zn inside Γ, then Γ f(z)dz = 2πinj=1 Res(zj ) Ch.6: Residue Theory 6.2 Trigonometric Integrals Over (0, 2π) Trigonometric Integrals Over [0, 2π] Our goal of this section is to apply the residue theory to evaluate real integrals of the form 2π 0 U(cos θ,sin θ)dθ (1) We use z = eiθ (0 ≤ θ ≤ 2π) to parameterize the closed positively oriented contour |z| = 1. Then a contour integral can be transformed into a real integral According to Euler’s equation, we have cos θ = eiθ + eiθ /2=(z + z−1)/2 sin θ = eiθ − eiθ /2i = (z − z−1)/2i Ch.6: Residue Theory 6.2 Trigonometric Integrals Over (0, 2π) Trigonometric Integrals Over [0, 2π] (Cont’d) Also taking dz = ieiθdθ = izdθ into account, Eq. (1) can be transformed into a complex contour integration as 2π 0 U(cos θ,sin θ)dθ = |z|=1 F(z)dz where the new integrand F is F(z) := U 12 z + 1z , 12i z − 1z · 1iz Ch.6: Residue Theory 6.2 Trigonometric Integrals Over (0, 2π) Trigonometric Integrals Over [0, 2π] (Cont’d) Of course, the function F must be a rational function of z Hence, it has only removable singularities (which can be ignored in evaluation integrals) or poles Consequently, by the residue theorem, our trigonometric integral equals 2πi time the sum of the residues at those poles of F that lie inside the unite circle

Ch.6:Residue Thoory Ch.6:Residue Theory L6.3 Improper Integrals of Certain Fumctions Over() L6.3 Improper Integral of Certain Functions Over ( Improper Integrals of Certain Functions Over(-oo,oo) Improper Integrals of Certain Functions Over (-oo,oo) (Cont'd) Given any function f continuous on(-oo,oo).the limit P in f(a)ds Lemma If f(z)=P(z)/Q(z)is the quotient of two polynomials such that is called the Cauchy principal value of the integral of f over (-oo,oo),and we write degree Q≥2+degree P p.v.f(r)dz :lim then We shall now show how the theory of residue can be used to lim f(z)dz=0 compute p.v.integrals for certain functions of f See Example 1 on page 319 to learn the basic idea of the where C is the upper half-circle of radius p defined in Eq.(4)on algorithm page 320 as shown in Figure 6.4 Ch.6:Residue Thoory Ch.6:Residue Theory 6.3 Improper Integrals of Certain Functions Over ( 6.4 Improper Integras involving Trigonometric Functions Improper Integrals of Certain Functions Over(-oo,oo) Improper Integrals Involving Trigonometric Functions (Cont'd) The purpose of this section is to use residue theory to evaluate integrals of the general forms: P(r) r P(r) Then the improper integralf()dr can be computed as p.v. /-Q(x) cosmz dr,p.v. JoQ(r) sinmz dx follo If we obtain the value of the integral pv.felt=m2ai∑(inide） 00 P()eimz dr Jo Q(E) the above two integrals can be obtained by computing the real and imaginary parts +口·811定+1意1意00

Ch.6: Residue Theory 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) Improper Integrals of Certain Functions Over (−∞, ∞) Given any function f continuous on (−∞,∞), the limit lim ρ→∞ ρ−ρ f(x)dx is called the Cauchy principal value of the integral of f over (−∞,∞), and we write p.v. ∞−∞ f(x)dx := limρ→∞ ρ−ρ f(x)dx We shall now show how the theory of residue can be used to compute p.v. integrals for certain functions of f See Example 1 on page 319 to learn the basic idea of the algorithm Ch.6: Residue Theory 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) Improper Integrals of Certain Functions Over (−∞, ∞) (Cont’d) Lemma If f(z) = P(z)/Q(z) is the quotient of two polynomials such that degree Q ≥ 2 + degree P then lim ρ→∞ C+ρ f(z)dz = 0 where C+ρ is the upper half-circle of radius ρ defined in Eq. (4) on page 320 as shown in Figure 6.4 Ch.6: Residue Theory 6.3 Improper Integrals of Certain Functions Over (−∞, ∞) Improper Integrals of Certain Functions Over (−∞, ∞) (Cont’d) Then the improper integral  ∞−∞ f(x)dx can be computed as follows p.v. ∞−∞ f(x)dx = limρ→∞ 2πi(residues inside Γρ) Ch.6: Residue Theory 6.4 Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions The purpose of this section is to use residue theory to evaluate integrals of the general forms: p.v. ∞−∞ P(x) Q(x) cos mx dx, p.v. ∞−∞ P(x) Q(x) sin mx dx If we obtain the value of the integral ∞−∞ P(x) Q(x)eimx dx the above two integrals can be obtained by computing the real and imaginary parts

Ch.6:Residue Thoory Ch.6:Residue Theory 6.4Imprope Integrals Involving Trigonone Functions Improperntegra Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions (Cont'd) (Cont'd) Lemma Then the improper integralf()dr can be computed as If m>0 and P/Q is the quotient of two polynomials such that follows degree Q≥1+degree P 广em号女=▣2a∑(e Q(z) then Thus mf eims P(d cos m Q( Q() P-c+Q(r) where C is the upper half-circle of radius p = sin m Q() 白·0+之。·急，是20C

Ch.6: Residue Theory 6.4 Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions (Cont’d) Lemma If m > 0 and P/Q is the quotient of two polynomials such that degree Q ≥ 1 + degree P then lim ρ→∞ C+ρ eimxP(x) Q(x)dz = 0 where C+ρ is the upper half-circle of radius ρ Ch.6: Residue Theory 6.4 Improper Integrals Involving Trigonometric Functions Improper Integrals Involving Trigonometric Functions (Cont’d) Then the improper integral  ∞−∞ f(x)dx can be computed as follows p.v. ∞−∞ eimxP(x) Q(x)dx = limρ→∞ 2πi(residues inside Γρ) Thus p.v. ∞−∞ cos mxP(x) Q(x)dx =  p.v. ∞−∞ eimxP(x) Q(x)dx p.v. ∞−∞ sin mxP(x) Q(x)dx =  p.v. ∞−∞ eimxP(x) Q(x)dx