Download - Integration in the complex plane
ntegration in the Complex Plane
Property of Amit Amola. Should be used for reference and with consent.
1
Prepared by:Amit AmolaSBSC(DU)
Defining Line Integrals in the Complex Plane
0a z
Nb z
1z2z
1
3z2
3
N
1Nz
nzn
x
y#. π ππ πΆ πππ‘π€πππ π§πβ1 πππ π§π
#. πΆπππ ππππ π‘βπ π π’ππ πΌπ = π=1
π π(π)(π§π β π§πβ1)
#. πΏππ‘ π‘βπ ππ’ππππ ππ π π’ππππ£ππ ππππ π β β,π π’πβ π‘βππ‘ βπ§π= π§π β π§πβ1 β 0 πππ ππππππ
πΌ =
π
π
π π§ ππ§ = limπββ
πΌπ
= limπββ
π=1
π
π(π)(π§π β π§πβ1)
(πππ π’ππ‘ ππ πππππππππππ‘ ππ π‘βπ πππ‘ππππ ππ π‘βπ πππ‘β π π’ππππ£ππ πππ)
C
Property of Amit Amola. Should be used for reference and with consent.
2
Equivalence Between Complex and Real Line Integrals
Note that-
So the complex line integral is equivalent to two real line integrals on C.
Property of Amit Amola. Should be used for reference and with consent.
3
Review of Line Integral Evaluation
t
t
-a a t-a a
1t2t
3t
1Nt C
nt
x
y
0t
f Nt t
1t 2t 3t 1Nt ntβ¦ β¦
0t f Nt tt
A line integral written as
is really a shorthand for:
where t is some parameterization of C :
or
Example- Parameterization of the circle x2 + y2 = a2
1) x= a cos(t), y= a sin(t), 0 t(=) 2
2) x= t, y= π2 β π‘2, t0 = π, tf = βa
And x= t, y=β π2 β π‘2, t0 = βπ, tf =a}
(π π , π π )
Property of Amit Amola. Should be used for reference and with consent.
4
Review of Line Integral Evaluation, contβd
1t2t
3t
1Nt C
nt
x
y
0t
f Nt t
While it may be possible to parameterize C(piecewise) using x and/or y as the independent parameter, it must be remembered that the other variable is always a function of that parameter i.e.-
(x is the independent parameter)
(y is the independent parameter)
(π π , π π )
Property of Amit Amola. Should be used for reference and with consent.
5
Line Integral ExampleConsider-
x
y
C
rz z
Itβs a useful result and a special case of the βresidue theoremβ
π. πΈπ£πππ’ππ‘π
π
π§πππ§ , π€βπππ
πΆ: π₯ = ππππ π, π¦ = ππ πππ, 0 β€ π β€ 2π
βΉ π§ = ππππ π + πππ πππ = ππππ ,βΉ ππ§ = πππππ ππ
Property of Amit Amola. Should be used for reference and with consent.
6
Cauchyβs Theorem
x
y
C
#. Cauchyβ²s Theorem: If π z is analytic in R then
πΆ
π π§ ππ§ = 0
#. First, note that if π π§ = π€ = π’ + ππ£, then
πΆ
π π§ ππ§ =
πΆ
(π’ππ₯ β π£ππ¦) + π
πΆ
π£ππ₯ + π’ππ¦ ;
β¦..πππ€ π€π π€πππ π’π π π π€πππ β ππππ€π π£πππ‘ππ πππππ¦π ππ πππ π’ππ‘ π‘π ππππ£π ππ’π π‘βπππππ:
R- a simply connected region
Construst vectors π = π’ π β π£ π,β¦π© = π£ π + π’ π , dr = ππ₯ π + ππ¦ π ,in the π₯π¦ plane and write the above equation as:
πΆ
π(π§) ππ§ =
πΆ
π¨. ππ + π
πΆ
π©. ππ =
πππ‘πππππ ππ πΆ
π» Γ π¨. πππ + π
πππ‘πππππ ππ πΆ
π» Γ π©. πππ, ππ’π‘
π. π» Γ π¨ = π.
π π ππ
ππ₯
π
ππ¦
π
ππ§π’ βπ£ 0
= βππ£
ππ₯βππ’
ππ¦= 0, π. π» Γ π© = π.
π π ππ
ππ₯
π
ππ¦
π
ππ§π£ π’ 0
=ππ’
ππ₯βππ£
ππ¦= 0
βΉ
πΆ
π π§ ππ§ = 0
Stokeβstheorem
C.R. Condition
C.R. ConditionProperty of Amit Amola. Should be used
for reference and with consent.7
Cauchyβs Theorem(contβd)
x
y
C
Consider R- a simply connected region
#. Cauchyβ²s Theorem:
If π z is analytic in R then
πΆ
π π§ ππ§ = 0
#. The proof using Stokeβ²s theorem requires that u x, y , v x, y havecontinuous first derivatives and that C be smoothβ¦
#. But the Goursat proof removes these restrictions; hence the theoremis often called the Cauchy β Goursat theorem
Property of Amit Amola. Should be used for reference and with consent.
8
Extension of Cauchyβs Theorem to Multiply-Connected Regions
x
y
1C2C
1c2c
x
y
1C2C
R- a multiply connected region
Rβ- a simply connected region
#. If π π§ is analytic in R then:
πΆ1,2
π π§ ππ§ =?
#. Introduce an infinitesimal- width βbridgeβ to make R into a simply connected region Rβ
πΆ1βπΆ2βπ1+π2
π π§ ππ§ =
πΆ1
π π§ ππ§ β
πΆ2
π π§ ππ§ = 0
#. It shows that integrals are independent of path.
(since integrals along π1 and π2 are in opposite directions and thus cancel each other)
Property of Amit Amola. Should be used for reference and with consent.
9
Fundamental Theorem of the Calculus of Complex Variables
az
bz
1z2z 3z
1Nz C
nz
x
y
1z
2z3z
Nz
π§π
π§π
π π§ ππ§ is a path independent in a simply connected region R
π§π
π§π
π π§ ππ§ = limπββ
π=1
π
π π βπ§π , βπ§π = π§π β π§πβ1,
π on C between znβ1 and zn
Suppose we can find πΉ z such that πΉβ² z =ππΉ
ππ§= π π§ ; hence π π β
ΞπΉπΞπ§π
=πΉ π§π β πΉ(π§πβ1)
βπ§π
= πΉ π§π β πΉ π§π (path independent if π π§ is analytic on the paths from π§π π‘π π§π)
β
Property of Amit Amola. Should be used for reference and with consent.
10
Fundamental Theorem of the Calculus of Complex Variables (contβd)
#. Fundamental Theorem of Calculus:
Suppose we can find πΉ π§ such that πΉβ² π§ =ππΉ
ππ§= π π§ :
βΉ
π§π
π§π
π π§ ππ§ =
π§π
π§πππΉ
ππ§ππ§ = πΉ π§π β πΉ π§π
#. This permits us to define indefinite integrals:
π§πππ§ =π§π+1
π + 1, sin π§ ππ§ = βcos π§, πππ§ππ§ =
πππ§
π, ππ‘π
#. In general
F(z)=
π§0
π§
π π for arbitrary π§0
#. All the indefinite integrals differ by only a constant.
Property of Amit Amola. Should be used for reference and with consent.
11
Cauchy Integral Formula
x
y
C
R
0z0C
1c2c
x
y
C
R- a simply connected region
#. π π§ is assumed analytic in R but we multiply by a factor1
π§ β π§0that is analytic
except at π§0 and consider the integral around πΆ
πΆ
π(π§)
(π§ β π§0)ππ§
#. To evaluate, consider the path πΆ + π1 + π2 + πΆ0 shown that encloses a simply βconnected region for which the integrand is analytic on and inside the path:
πΆ+π1+π2+πΆ0
π(π§)
(π§ β π§0)ππ§ = 0 βΉ
πΆ
π(π§)
(π§ β π§0)ππ§ = β
πΆ0
π(π§)
(π§ β π§0)ππ§
Property of Amit Amola. Should be used for reference and with consent.
12
Cauchy Integral Formula, contβd
00 0C C
f z f zdz dz
z z z z
0z
0C
C
Evaluate the πΆ0 integral on a circular path,
for r βΆ 0
β βCauchy Integral Formula
The value of π π§ at π§0 is completely determined by its values on πΆ!Property of Amit Amola. Should be used for reference and with consent.
13
Cauchy Integral Formula, contβd
0z
0C
C
0z
C
Note that if π§0 is outside C, the integrand is analytic inside C, hence by the Cauchy Integral Theorem,
1
2ππ
πΆ
π(π§)
π§ β π§0ππ§ = 0
In summary,
πΆ
π(π§)
π§ β π§0ππ§ =
2ππ π π§0 , π§0 inside πΆ0, π§0 outside πΆ
Property of Amit Amola. Should be used for reference and with consent.
14
Derivative Formula0z
C
Since π(π§) is analytic in C , its derivative exists;letβs express it in terms of the Cauchy Formula:
π π§0 =1
2ππ
πΆ
π(π§)
π§ β π§0ππ§
.So
β
β
β Weβve also just proved we can differentiate w.r.t. π§0 under the integral sign!Property of Amit Amola. Should be used
for reference and with consent.15
Derivative Formulas, contβd
0z
C
If π(π§) is analytic in C, then its derivatives of all orders exist, and hence they are analytic as well.
In general,
Similarly
or
βProperty of Amit Amola. Should be used for reference and with consent.
16