cvt sajid ali complex integration complex integration...
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CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Sajid Ali
SEECS-NUST
October 2, 2017
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
A complex line integral is given by∫Cf (z)dz ,
where C is the curve along which the integration is carriedout.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
A complex line integral is given by∫Cf (z)dz ,
where C is the curve along which the integration is carriedout. In order to evaluate it we need to characterize domains.
Simply Connected Domains: A domain D is calledsimply connected if every closed path in it can becontinuously deformed (or shrinked) to only points of D.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
A complex line integral is given by∫Cf (z)dz ,
where C is the curve along which the integration is carriedout. In order to evaluate it we need to characterize domains.
Simply Connected Domains: A domain D is calledsimply connected if every closed path in it can becontinuously deformed (or shrinked) to only points of D.
For example, complex plane, interior of a an open disk, etc.Counter example, a unit circle without origin, an annulusregion etc..
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
How to evaluate complex line integral:Way-1: Direct substitution by using fundamental theorem ofcomplex analysis
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
How to evaluate complex line integral:Way-1: Direct substitution by using fundamental theorem ofcomplex analysisWay-2: Identify path and then carry out integration
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-1: Suppose F is analytic in a simply connected domainD then for all paths between z0 and z1 the fundamentaltheorem of complex line integrals states∫ z1
z0
F ′(z)dz = F (z1)− F (z0)
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-1: Suppose F is analytic in a simply connected domainD then for all paths between z0 and z1 the fundamentaltheorem of complex line integrals states∫ z1
z0
F ′(z)dz = F (z1)− F (z0)
Example-1: ∫ 1+i
0z2dz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-1: Suppose F is analytic in a simply connected domainD then for all paths between z0 and z1 the fundamentaltheorem of complex line integrals states∫ z1
z0
F ′(z)dz = F (z1)− F (z0)
Example-1: ∫ 1+i
0z2dz =?
Since the domain of f (z) = z2 is the entire complex planewhich is simply connected and the function is analytic in itand
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-1: Suppose F is analytic in a simply connected domainD then for all paths between z0 and z1 the fundamentaltheorem of complex line integrals states∫ z1
z0
F ′(z)dz = F (z1)− F (z0)
Example-1: ∫ 1+i
0z2dz =?
Since the domain of f (z) = z2 is the entire complex planewhich is simply connected and the function is analytic in itand
d
dz
z3
3= z2
⇒∫ 1+i
0z2dz =
z3
3|1+i0
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-1: Suppose F is analytic in a simply connected domainD then for all paths between z0 and z1 the fundamentaltheorem of complex line integrals states∫ z1
z0
F ′(z)dz = F (z1)− F (z0)
Practice:
1.
∫ πi
−πicos zdz =?
2.
∫ i
−i
1
zdz =?.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-2: Parameterize the curve
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-2: Parameterize the curveLet C be a piecewise smooth path, represented by z = z(t),where a < t < b. Let f (z) be a continuous function on Cthen ∫
cf (z)dz =
∫ b
af (z)z(t)dt
where z = x + i y .
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Way-2: Parameterize the curveLet C be a piecewise smooth path, represented by z = z(t),where a < t < b. Let f (z) be a continuous function on Cthen ∫
cf (z)dz =
∫ b
af (z)z(t)dt
where z = x + i y .
Examples:For boundary of a unit circle C evaluate the integral∫
C
1
zdz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Examples:For boundary of a unit circle C evaluate the integral∫
C
1
zdz =?
Step-1 As C is a unit circle so parameterize
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Examples:For boundary of a unit circle C evaluate the integral∫
C
1
zdz =?
Step-1 As C is a unit circle so parameterize
z(t) = cos t + i sin t = e it
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Examples:For boundary of a unit circle C evaluate the integral∫
C
1
zdz =?
Step-1 As C is a unit circle so parameterize
z(t) = cos t + i sin t = e it
Step-2 Differentiate z = ie it
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Examples:For boundary of a unit circle C evaluate the integral∫
C
1
zdz =?
Step-1 As C is a unit circle so parameterize
z(t) = cos t + i sin t = e it
Step-2 Differentiate z = ie it
Step-3 Calculate f (z(t)) = e−it
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Examples:For boundary of a unit circle C evaluate the integral∫
C
1
zdz =?
Step-1 As C is a unit circle so parameterize
z(t) = cos t + i sin t = e it
Step-2 Differentiate z = ie it
Step-3 Calculate f (z(t)) = e−it
Step-4 Evaluate the integral∫Cf (z)dz =
∫ 2π
0ie−ite itdt
= 2πi Ans.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Follow-up Questions:What ifQ-1 C is a rectangular box with z = 0 at center?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Follow-up Questions:What ifQ-1 C is a rectangular box with z = 0 at center?Q-2 C is a closed curve with z = 0 not at its center?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Follow-up Questions:What ifQ-1 C is a rectangular box with z = 0 at center?Q-2 C is a closed curve with z = 0 not at its center?Q-3 C is a closed curve and does not contain z = 0?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Follow-up Questions:What ifQ-1 C is a rectangular box with z = 0 at center?Q-2 C is a closed curve with z = 0 not at its center?Q-3 C is a closed curve and does not contain z = 0?Q-4 C is a circle and contain z = 1, so∫
C
1
z − 1dz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Follow-up Questions:What ifQ-1 C is a rectangular box with z = 0 at center?Q-2 C is a closed curve with z = 0 not at its center?Q-3 C is a closed curve and does not contain z = 0?Q-4 C is a circle and contain z = 1, so∫
C
1
z − 1dz =?
Q-5 C is a unit circle∫C
(z + z−1)dz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Practice:Evaluate the same integral
∫C 1/zdz over more paths !!!
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Practice:Evaluate the same integral
∫C 1/zdz over more paths !!!
1. For an ellipse
z(t) = a cos t + ib sin t
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integration
Practice:Evaluate the same integral
∫C 1/zdz over more paths !!!
1. For an ellipse
z(t) = a cos t + ib sin t
2. For a closed parabolic path
z(t) = t + it2, 0 < t < 1
z(t) = 1− t + i(1− t), 0 < t < 1
evaluate the integral ∫Cz2 dz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
We have observed that the answer to a complex line integralalong a closed curve is zero or 2πi that is whether singularityis outside or inside the closed curve. It is the consequence ofa famous theorem in complex analysis.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
We have observed that the answer to a complex line integralalong a closed curve is zero or 2πi that is whether singularityis outside or inside the closed curve. It is the consequence ofa famous theorem in complex analysis.
Cauchy Integral Theorem:If f (z) is analytic in a simply connected domain D, then forevery closed path C in D∫
Cf (z)dz = 0
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
We have observed that the answer to a complex line integralalong a closed curve is zero or 2πi that is whether singularityis outside or inside the closed curve. It is the consequence ofa famous theorem in complex analysis.
Cauchy Integral Theorem:If f (z) is analytic in a simply connected domain D, then forevery closed path C in D∫
Cf (z)dz = 0
Practice: For an arbitrary closed curve C
(i)
∫Cezdz =?, (ii)
∫C
cos(z)dz =?, (iii)
∫Czndz =?,
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
We have observed that the answer to a complex line integralalong a closed curve is zero or 2πi that is whether singularityis outside or inside the closed curve. It is the consequence ofa famous theorem in complex analysis.
Cauchy Integral Theorem:If f (z) is analytic in a simply connected domain D, then forevery closed path C in D∫
Cf (z)dz = 0
Practice: For an arbitrary closed curve C
(i)
∫Cezdz = 0, (ii)
∫C
cos(z)dz = 0, (iii)
∫Czndz = 0,
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
We have observed that the answer to a complex line integralalong a closed curve is zero or 2πi that is whether singularityis outside or inside the closed curve. It is the consequence ofa famous theorem in complex analysis.
Cauchy Integral Theorem:If f (z) is analytic in a simply connected domain D, then forevery closed path C in D∫
Cf (z)dz = 0
Practice: For an arbitrary closed curve C∫C
sec (z)dz = 0,
∫C
1
z2 + 4dz = 0, C = Unit Circle
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic? NO !!!
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic? NO !!! For example
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic? NO !!! For example∫
C
1
z2dz = 0 C = Unit Circle
where f (z) = 1/z2 is clearly not analytic at z = 0.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic? NO !!! For example∫
C
1
z2dz = 0 C = Unit Circle
where f (z) = 1/z2 is clearly not analytic at z = 0.Therefore Cauchy integral theorem only provide necessarycondition and not the sufficient condition.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic? NO !!! For example∫
C
1
z2dz = 0 C = Unit Circle
where f (z) = 1/z2 is clearly not analytic at z = 0.Therefore Cauchy integral theorem only provide necessarycondition and not the sufficient condition.
Independence of path:If f (z) is analytic in a simply connected domain D, then theintegral of f is independent of path in D.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Theorem
What about the converse of Cauchy integral theorem? i.e., ifthe integral of a complex function is zero does it guaranteethat the function is analytic? NO !!! For example∫
C
1
z2dz = 0 C = Unit Circle
where f (z) = 1/z2 is clearly not analytic at z = 0.Therefore Cauchy integral theorem only provide necessarycondition and not the sufficient condition.
Independence of path:If f (z) is analytic in a simply connected domain D, then theintegral of f is independent of path in D.This is an important result from the physical point of view.If f (z) has the sense of a vector field on the plane thenabove theorem implies that the vector field is conservative.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
If f (z) is analytic in a simply connected domain D, then∫C
f (z)
z − z0dz = 2πif (z0)
where C is taken counterclockwise and contains z0.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
If f (z) is analytic in a simply connected domain D, then∫C
f (z)
z − z0dz = 2πif (z0)
where C is taken counterclockwise and contains z0.
Example-1:Suppose C : any contour containing z0 = 2∫
c
ez
z − 2dz =
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
If f (z) is analytic in a simply connected domain D, then∫C
f (z)
z − z0dz = 2πif (z0)
where C is taken counterclockwise and contains z0.
Example-1:Suppose C : any contour containing z0 = 2∫
c
ez
z − 2dz = 2πi |ez |z=2
= 2πiez |z=2, (as ez is analytic)
= 2πie2
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
PracticeSuppose C : is the contours (i) |z | = 1 (ii) |z − 1| = 2 (iii)|z + 1| = 2. Evaluate ∫
c
1
z2 − 4dz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-2:Suppose we have C : |z − 1| = 1
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-2:Suppose we have C : |z − 1| = 1 then find the value of
I =
∫C
z2 + 1
z2 − 1dz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-2:Suppose we have C : |z − 1| = 1 then find the value of
I =
∫C
z2 + 1
z2 − 1dz =?
Sol. Since C represent a circle of radius one centered at oneunit on the x-axis therefore it includes the point ofsingularity of f (z).
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-2:Suppose we have C : |z − 1| = 1 then find the value of
I =
∫C
z2 + 1
z2 − 1dz =?
Sol. Since C represent a circle of radius one centered at oneunit on the x-axis therefore it includes the point ofsingularity of f (z). But we can write the integrand such that
I =
∫C
z2+1z+1
z − 1dz
in which case ∴ f (z) = z2+1z+1 is analytic on the given
contour, C , as z = −1 is not on C so, f (1) = 1+11+1 = 1
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-2:Suppose we have C : |z − 1| = 1 then find the value of
I =
∫C
z2 + 1
z2 − 1dz =?
Sol. Since C represent a circle of radius one centered at oneunit on the x-axis therefore it includes the point ofsingularity of f (z). But we can write the integrand such that
I =
∫C
z2+1z+1
z − 1dz
in which case ∴ f (z) = z2+1z+1 is analytic on the given contour,
C , as z = −1 is not on C so, f (1) = 1+11+1 = 1 therefore∫
C= 2πi
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-3:Suppose we have C : |z + 1| = 1 then find the value of
I =
∫C
z2 + 1
z2 − 1dz =?
Sol. Since C represent a circle of radius one centered at oneunit on the x-axis therefore it includes the point ofsingularity of f (z).
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Complex Integral Formula
Example-3:Suppose we have C : |z + 1| = 1 then find the value of
I =
∫C
z2 + 1
z2 − 1dz =?
Sol. Since C represent a circle of radius one centered at oneunit on the x-axis therefore it includes the point ofsingularity of f (z). But we can write the integrand such that
I =
∫C
z2+1z−1z + 1
dz
in which case ∴ f (z) = z2+1z−1 is analytic on the given contour,
C , as z = 1 is not on C so, f (−1) = 2−2 = −1 therefore∫
C= −2πi
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Laurent Series
Review:The coefficients an in the Laurent series are determined froma complex line integral
an =1
2πi
∫C
f (z)
(z − z0)n+1dz
where C is a counter-clock wise oriented closed curve lyingin an annulus in which f (z) is analytic. The expansion off (z) is then valid in the annulus.
Note that the power n = −1, yields the complex integral ofthe original function
a(−1) =1
2πi
∫Cf (z) dz
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
The coefficient of the term 1/z in the Laurent series of acomplex function f (z) is known as the residue of f (z) atz = z0,
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
The coefficient of the term 1/z in the Laurent series of acomplex function f (z) is known as the residue of f (z) atz = z0, i.e.,
Resz=z0
(f (z)) = a(−1).
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
The coefficient of the term 1/z in the Laurent series of acomplex function f (z) is known as the residue of f (z) atz = z0, i.e.,
Resz=z0
(f (z)) = a(−1).
Therefore the integral of f (z) over C , a counter-clock wiseoriented closed curve lying in an annulus in which f (z) isanalytic ∫
Cf (z) dz = 2πi Res
z=z0(f (z)).
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-1:Integrate f (z) = z−4 sin z around unit circle.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-1:Integrate f (z) = z−4 sin z around unit circle.Solution:The Laurent series of f (z) is
f (z) = z−4(z − z3
3!+
z5
5!− ...
)=
1
z3− 1
3!z+
z
5!− ...
Therefore,
Resz=z0
(f (z)) = −1
6.
Hence the integral is∫Cf (z) dz = −2πi × 1
6= −πi
3.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
How to compute residues?For a function of the form
f (z) =p(z)
q(z)
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
How to compute residues?For a function of the form
f (z) =p(z)
q(z)
such that p(z0) 6= 0, q(z0) = 0, q′(z0) 6= 0, then the residueof f (z) at z = z0 is given by
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
How to compute residues?For a function of the form
f (z) =p(z)
q(z)
such that p(z0) 6= 0, q(z0) = 0, q′(z0) 6= 0, then the residueof f (z) at z = z0 is given by
Resz=z0
(f (z)) =p(z0)
q′(z0).
Note that q(z) has a simple zero at z = z0.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
How to compute residues?For a function of the form
f (z) =p(z)
q(z)
such that p(z0) 6= 0, q(z0) = 0, q′(z0) 6= 0, then the residueof f (z) at z = z0 is given by
Resz=z0
(f (z)) =p(z0)
q′(z0).
Note that q(z) has a simple zero at z = z0.
What if q(z) has a zero of order n?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-11. For a unit circle centered at origin, compute∫
C
1
zdz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-11. For a unit circle centered at origin, compute∫
C
1
zdz =?
Solution:Here q(z) = z which has a simple zero at z = 0 therefore
Resz=0
(f (z)) =p(0)
q′(0)=
1
1= 1.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-11. For a unit circle centered at origin, compute∫
C
1
zdz =?
Solution:Here q(z) = z which has a simple zero at z = 0 therefore
Resz=0
(f (z)) =p(0)
q′(0)=
1
1= 1.
Therefore, ∫C
1
zdz = 2πi re-verified
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-2:2. For a unit circle centered at z = 1, compute∫
C
sin z
cos zdz =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-2:2. For a unit circle centered at z = 1, compute∫
C
sin z
cos zdz =?
Solution:Here q(z) = cos z which has simple zeroes atz = ±π/2,±3π/2, ... but there is only one isolatedsingularity inside C therefore
Resz=π/2
(f (z)) =p(π/2)
q′(π/2)=
1
−1= −1.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-2:2. For a unit circle centered at z = 1, compute∫
C
sin z
cos zdz =?
Solution:Here q(z) = cos z which has simple zeroes atz = ±π/2,±3π/2, ... but there is only one isolatedsingularity inside C therefore
Resz=π/2
(f (z)) =p(π/2)
q′(π/2)=
1
−1= −1.
Therefore, ∫C
1
zdz = −2πi
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Practice:3. What about residues of
f (z) =sin z
cos z
at z = ±π/2,±3π/2, ...?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Practice:4. Find the residues of
f (z) =1
z2 sin z
z =?
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Multiple Residues:If a complex function f (z) is analytic everywhere in domainD except for finitely many points z1, z2, ..., zn then for aclosed simple curve C enclosing all singular points Cauchyresidue theorem states
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Multiple Residues:If a complex function f (z) is analytic everywhere in domainD except for finitely many points z1, z2, ..., zn then for aclosed simple curve C enclosing all singular points Cauchyresidue theorem states∫
Cf (z)dz = 2πi
∑k=1
Resz=zk
(f (z)).
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-1:1. Find the integral ∫
C
1
z2 − 5z + 6
over C which is a (i) circle of radius 4, (ii) circle of radius5/2.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-1:1. Find the integral ∫
C
1
z2 − 5z + 6
over C which is a (i) circle of radius 4, (ii) circle of radius5/2.Solution (i):Here q(z) = z2 − 5z + 6 which has simple zeroes at z = 2, 3and both isolated singularities are inside C therefore
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-1:1. Find the integral ∫
C
1
z2 − 5z + 6
over C which is a (i) circle of radius 4, (ii) circle of radius5/2.Solution (i):Here q(z) = z2 − 5z + 6 which has simple zeroes at z = 2, 3and both isolated singularities are inside C therefore
Resz=2
(f (z)) =p(2)
q′(2)=
1
−1= −1.
Resz=3
(f (z)) =p(3)
q′(3)=
1
1= 1.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Example-1:1. Find the integral ∫
C
1
z2 − 5z + 6
over C which is a (i) circle of radius 4, (ii) circle of radius5/2.Solution (ii):Since the radius of other circle is 5/2, which does notinclude the singularity 3 therefore∫
C
1
zdz = 2πi
(Resz=2
(f (z)))
= 2πi
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Practice:2. Find the integral ∫
C
eaz
z2 + 1, a ∈ R
over C which is a circle of radius 2.
3. Find the integral ∫C
tan z
z2 − 1, a ∈ R
over C which is a circle of radius 3/2.
CVT
Sajid Ali
ComplexIntegration
Simply ConnectedDomains
Complex LineIntegrals
Cauchy IntegralTheorem
Cauchy IntegralFormula
Residues
Residues
Practice:4. Evaluate the following integral, where C is the ellipse9x2 + y2 = 9 ∫
C
(zeπz
z4 − 16+ zeπ/z
).
5. Evaluate the following integral, where C is a unit circle atorigin ∫
C
cosh z
z2 − 3izdz .