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#. 𝑛 𝑜𝑛 𝐶 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑧𝑛−1 𝑎𝑛𝑑 𝑧𝑛

#. 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡ℎ𝑒 𝑠𝑢𝑚𝑠𝐼𝑛 = 𝑛=1

𝑁 𝑓(𝑛)(𝑧𝑛 − 𝑧𝑛−1)

#. 𝐿𝑒𝑡 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑏𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛𝑠 𝑁 → ∞,𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∆𝑧𝑛= 𝑧𝑛 − 𝑧𝑛−1 → 0 𝑎𝑛𝑑 𝑑𝑒𝑓𝑖𝑛𝑒

𝐼 =

𝑎

𝑏

𝑓 𝑧 𝑑𝑧 = lim𝑁→∞

𝐼𝑁

= lim𝑁→∞

𝑛=1

𝑁

𝑓(𝑛)(𝑧𝑛 − 𝑧𝑛−1)

(𝑟𝑒𝑠𝑢𝑙𝑡 𝑖𝑠 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑡𝑎𝑖𝑙𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ 𝑠𝑢𝑏𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛)

C


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Equivalence Between Complex and Real Line Integrals

Note that-

So the complex line integral is equivalent to two real line integrals on C.


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 nt… …

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= 𝑎2 − 𝑡2, t0 = 𝑎, tf = −a

And x= t, y=− 𝑎2 − 𝑡2, t0 = −𝑎, tf =a}

(𝒙 𝒕 , 𝒚 𝒕 )


4

Review of Line Integral Evaluation, cont’d

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)

(𝒙 𝒕 , 𝒚 𝒕 )


5

Line Integral ExampleConsider-

x

y

C

rz z

It’s a useful result and a special case of the “residue theorem”

𝑄. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒

𝑐

𝑧𝑛𝑑𝑧 , 𝑤ℎ𝑒𝑟𝑒

𝐶: 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃, 0 ≤ 𝜃 ≤ 2𝜋

⟹ 𝑧 = 𝑟𝑐𝑜𝑠𝜃 + 𝑖𝑟𝑠𝑖𝑛𝜃 = 𝑟𝑒𝑖𝜃 ,⟹ 𝑑𝑧 = 𝑟𝑖𝑒𝑖𝜃 𝑑𝜃


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Cauchy’s Theorem

x

y

C

#. Cauchy′s Theorem: If 𝑓 z is analytic in R then

𝐶

𝑓 𝑧 𝑑𝑧 = 0

#. First, note that if 𝑓 𝑧 = 𝑤 = 𝑢 + 𝑖𝑣, then

𝐶

𝑓 𝑧 𝑑𝑧 =

𝐶

(𝑢𝑑𝑥 − 𝑣𝑑𝑦) + 𝑖

𝐶

𝑣𝑑𝑥 + 𝑢𝑑𝑦 ;

…..𝑛𝑜𝑤 𝑤𝑒 𝑤𝑖𝑙𝑙 𝑢𝑠𝑒 𝑎 𝑤𝑒𝑙𝑙 − 𝑘𝑛𝑜𝑤𝑛 𝑣𝑒𝑐𝑡𝑜𝑟 𝑎𝑛𝑎𝑙𝑦𝑠𝑖𝑠 𝑟𝑒𝑠𝑢𝑙𝑡 𝑡𝑜 𝑝𝑟𝑜𝑣𝑒 𝑜𝑢𝑟 𝑡ℎ𝑒𝑜𝑟𝑒𝑚:

R- a simply connected region

Construst vectors 𝐀 = 𝑢 𝒙 − 𝑣 𝒚,…𝑩 = 𝑣 𝒙 + 𝑢 𝒚 , dr = 𝑑𝑥 𝒙 + 𝑑𝑦 𝒚 ,in the 𝑥𝑦 plane and write the above equation as:

𝐶

𝑓(𝑧) 𝑑𝑧 =

𝐶

𝑨. 𝑑𝒓 + 𝑖

𝐶

𝑩. 𝑑𝒓 =

𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑜𝑓 𝐶

𝛻 × 𝑨. 𝒛𝑑𝑆 + 𝑖

𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑜𝑓 𝐶

𝛻 × 𝑩. 𝒛𝑑𝑆, 𝑏𝑢𝑡

𝒛. 𝛻 × 𝑨 = 𝒛.

𝒙 𝒚 𝒛𝜕

𝜕𝑥

𝜕

𝜕𝑦

𝜕

𝜕𝑧𝑢 −𝑣 0

= −𝜕𝑣

𝜕𝑥−𝜕𝑢

𝜕𝑦= 0, 𝒛. 𝛻 × 𝑩 = 𝒛.

𝒙 𝒚 𝒛𝜕

𝜕𝑥

𝜕

𝜕𝑦

𝜕

𝜕𝑧𝑣 𝑢 0

=𝜕𝑢

𝜕𝑥−𝜕𝑣

𝜕𝑦= 0

⟹

𝐶


Stoke’stheorem

C.R. Condition

C.R. ConditionProperty of Amit Amola. Should be used

for reference and with consent.7

Cauchy’s Theorem(cont’d)

x

y

C

Consider R- a simply connected region

#. Cauchy′s Theorem:

If 𝑓 z is analytic in R then

𝐶


#. The proof using Stoke′s theorem requires that u x, y , v x, y havecontinuous first derivatives and that C be smooth…

#. But the Goursat proof removes these restrictions; hence the theoremis often called the Cauchy − Goursat theorem


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Extension of Cauchy’s Theorem to Multiply-Connected Regions

x

y

1C2C

1c2c

x

y

1C2C

R- a multiply connected region

R’- a simply connected region

#. If 𝑓 𝑧 is analytic in R then:

𝐶1,2

𝑓 𝑧 𝑑𝑧 =?

#. Introduce an infinitesimal- width “bridge” to make R into a simply connected region R’

𝐶1−𝐶2−𝑐1+𝑐2


𝐶1

𝑓 𝑧 𝑑𝑧 −

𝐶2


#. It shows that integrals are independent of path.

(since integrals along 𝑐1 and 𝑐2 are in opposite directions and thus cancel each other)


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Fundamental Theorem of the Calculus of Complex Variables

az

bz

1z2z 3z

1Nz C

nz

x

y

1z

2z3z

Nz

𝑧𝑎

𝑧𝑏

𝑓 𝑧 𝑑𝑧 is a path independent in a simply connected region R

𝑧𝑎

𝑧𝑏

𝑓 𝑧 𝑑𝑧 = lim𝑁→∞

𝑛=1

𝑁

𝑓 𝑛 ∆𝑧𝑛 , ∆𝑧𝑛 = 𝑧𝑛 − 𝑧𝑛−1,

𝑛 on C between zn−1 and zn

Suppose we can find 𝐹 z such that 𝐹′ z =𝑑𝐹

𝑑𝑧= 𝑓 𝑧 ; hence 𝑓 𝑛 ≈

Δ𝐹𝑛Δ𝑧𝑛

=𝐹 𝑧𝑛 − 𝐹(𝑧𝑛−1)

∆𝑧𝑛

= 𝐹 𝑧𝑏 − 𝐹 𝑧𝑎 (path independent if 𝑓 𝑧 is analytic on the paths from 𝑧𝑎 𝑡𝑜 𝑧𝑏)

⇒


10

Fundamental Theorem of the Calculus of Complex Variables (cont’d)

#. Fundamental Theorem of Calculus:

Suppose we can find 𝐹 𝑧 such that 𝐹′ 𝑧 =𝑑𝐹

𝑑𝑧= 𝑓 𝑧 :

⟹

𝑧𝑎

𝑧𝑏


𝑧𝑎

𝑧𝑏𝑑𝐹

𝑑𝑧𝑑𝑧 = 𝐹 𝑧𝑏 − 𝐹 𝑧𝑎

#. This permits us to define indefinite integrals:

𝑧𝑛𝑑𝑧 =𝑧𝑛+1

𝑛 + 1, sin 𝑧 𝑑𝑧 = −cos 𝑧, 𝑒𝑎𝑧𝑑𝑧 =

𝑒𝑎𝑧

𝑎, 𝑒𝑡𝑐

#. In general

F(z)=

𝑧0

𝑧

𝑓 𝑑 for arbitrary 𝑧0

#. All the indefinite integrals differ by only a constant.


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Cauchy Integral Formula

x

y

C

R

0z0C

1c2c

x

y

C

R- a simply connected region

#. 𝑓 𝑧 is assumed analytic in R but we multiply by a factor1

𝑧 − 𝑧0that is analytic

except at 𝑧0 and consider the integral around 𝐶

𝐶

𝑓(𝑧)

(𝑧 − 𝑧0)𝑑𝑧

#. To evaluate, consider the path 𝐶 + 𝑐1 + 𝑐2 + 𝐶0 shown that encloses a simply −connected region for which the integrand is analytic on and inside the path:

𝐶+𝑐1+𝑐2+𝐶0

𝑓(𝑧)

(𝑧 − 𝑧0)𝑑𝑧 = 0 ⟹

𝐶

𝑓(𝑧)

(𝑧 − 𝑧0)𝑑𝑧 = −

𝐶0

𝑓(𝑧)

(𝑧 − 𝑧0)𝑑𝑧


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Cauchy Integral Formula, cont’d

00 0C C

f z f zdz dz

z z z z

0z

0C

C

Evaluate the 𝐶0 integral on a circular path,

for r ⟶ 0

⇒ ⇒Cauchy Integral Formula

The value of 𝑓 𝑧 at 𝑧0 is completely determined by its values on 𝐶!Property of Amit Amola. Should be used for reference and with consent.

13

Cauchy Integral Formula, cont’d

0z

0C

C

0z

C

Note that if 𝑧0 is outside C, the integrand is analytic inside C, hence by the Cauchy Integral Theorem,

1

2𝜋𝑖

𝐶

𝑓(𝑧)

𝑧 − 𝑧0𝑑𝑧 = 0

In summary,

𝐶

𝑓(𝑧)

𝑧 − 𝑧0𝑑𝑧 =

2𝜋𝑖 𝑓 𝑧0 , 𝑧0 inside 𝐶0, 𝑧0 outside 𝐶


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Derivative Formula0z

C

Since 𝑓(𝑧) is analytic in C , its derivative exists;let’s express it in terms of the Cauchy Formula:

𝑓 𝑧0 =1

2𝜋𝑖

𝐶

𝑓(𝑧)

𝑧 − 𝑧0𝑑𝑧

.So

⇒

⇒

⇒ We’ve also just proved we can differentiate w.r.t. 𝑧0 under the integral sign!Property of Amit Amola. Should be used

for reference and with consent.15

Derivative Formulas, cont’d

0z

C

If 𝑓(𝑧) is analytic in C, then its derivatives of all orders exist, and hence they are analytic as well.

In general,

Similarly

or

⇒Property of Amit Amola. Should be used for reference and with consent.

16

integration in the complex plane

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