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Residue Integration

1. Complex power series

2. The Laurent series and the Residue Theorem

3. Integrals involving functions with branch points

Week 6

1. Complex power series

In what follows, we’ll need the following results:

Example 1:

.1lim,lim

n

nnn

c

a

dnc

bna

2

Then:

(a) If L < 1, the series converges absolutely.

(b) If L > 1, the series diverges.

(c) If L = 1, the test is inconclusive.

Complex series are similar to their real counterparts and ‘obey’ similar convergence theorems (the Comparison Test, Ratio Test, Root Test, etc.).

Theorem 1: Ratio Test (light version)

Let the terms of a series z1 + z2 + ... be such that the following limit exists:

.lim 1 Lz

z

n

n

n

Proof: see Kreyszig, Section 15.1

(1)

3

Then:

(a) If L < 1, the series it converges absolutely.

(b) If L > 1, the series diverges.

(c) If L = 1, the test is inconclusive.

Theorem 2: Root Test (light version)

Let the terms of a series z1 + z2 + ... be such that the following limit exists:

.lim Lznn

n

Proof: see Kreyszig, Section 15.1

Comment:

The heavy versions of the Ratio and Root Tests can be found in Kreyszig, section 15.1 (Theorems 7 and 9). They don’t require that limits (1)–(2) exist. [Give examples where they indeed don’t.]

(2)

4

We shall now review power series, i.e. the series of the form

,)(0

0

n

nn zza

where an are (complex) coefficients.

(3)

Theorem 3: Convergence of a power series

(a) Every power series (3) converges at the centre, i.e. at z = z0.

(b) If (3) converges at z1 ≠ z0, it also converges absolutely for

every z closer to z0 than z1 – i.e. if | z − z0 | < | z1 − z0 | (see the figure in the next slide).

(c) If (3) diverges at z2, it diverges for all z farther away from

z0 than z2.

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۞ The largest circle centred at z0, such that a power series (3) converges at all of its interior points (but not necessarily at the boundary), is called the circle of convergence. The circle’s radius is called the radius of convergence of (3).

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Example 2:

The geometric series,

,0

n

nzS

converges absolutely if | z | < 1 and diverges if | z | ≥ 1. Thus, its radius of convergence is equal to 1.

This series can be summed up, yielding

.1

1

zS

Comment:

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Example 3:

The series

0 !n

n

n

z

converges absolutely for all z. Thus, it has an infinite radius of convergence.

This series can be summed up, yielding

.ezS

Comment:

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Example 4:

The series

0

!n

nzn

diverges for all z ≠ 0. Thus, it has a zero radius of convergence.

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Theorem 4: Cauchy–Hadamard Formula

Let the coefficients of a power series be such that the following limit exists:

.lim 1 La

a

n

n

n

Then: (a) If L = 0, the series converges for all z.

(b) If ∞ > L > 0, the radius R of convergence of the series is

.1

LR

Proof: follows from the Ratio Test.

(4)

(c) If L = ∞, the series diverges for all z.

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Theorem 5: Uniqueness of power series

Let power series S1 and S2 be convergent and have equal

centres and equal sums for a circle | z | < R, where R > 0. Then the coefficients of these series are equal.

Hence, if a function is represented by a power series with a certain centre, this representation is unique.

Theorem 6:If a series has a zero sum in a circle of non-zero radius, the coefficients of this series are all zero.

Proof: follows from Theorem 5.

Proof: see Kreyszig, section 15.3 (non-examinable)

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Theorem 7: Term-wise differentiation/integration of power seriesLet S be a power series of form (3), such that limit (4) with a

finite L exists for its coefficients, and let SD and SI be the series obtained by term-wise differentiation and integration of S.

Then all three series have the same radius of convergence.

Hint: Apply the Ratio Test to SD and SI.

Proof:

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Theorem 8:

A power series with a nonzero radius of convergence represents an analytic function at all interior points of its circle of convergence.

The derivatives of this function can be obtained by term-wise differentiation of the original series.

The ‘derivative series’ have the same radius of convergence as the original series.

Thus, differentiability of a function in a domain = analyticity (infinite differentiability) of the function in the domain = the function’s representability by a power series in the domain.

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۞ Let f(z) be an analytic function in a domain including a point z0. Then the power series

,)(0

0

n

nn zza

with its coefficients given by

is called the Taylor series of f(z) at z0.

,d

d

!

1

0zzn

n

n z

f

na

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Theorem 9: Taylor’s Theorem

Let f(z) be analytic in a domain D, and let z0 be a point in D. Then there exists precisely one Taylor series with centre at z0 converging to f(z) in the largest open circle with centre z0 in

which f(z) is analytic.