(1) Power series

1
Section 6
SECTION 6 Power Series I - Taylor Series
(1) Power series (2) Convergence of Power series
and the Radius of Convergence (3) The
Cauchy-Hadamard formula (4) Taylors series
2
Why Series ?
Section 6
where C is
We can expand the integrand.
by Formula for derivatives (?? )
Analytic (?? )
3
Section 6
Can we expand all functions in series ?
We can expand analytic functions in
special series called Power Series
How do we find these series ?
(1) Using Taylors Theorem (2) Using other known
series (and other tricks)
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Power Series
Section 6
A series in powers of
e.g.
5
Power Series
Section 6
A series in powers of
centre
e.g.
6
Power Series
Section 6
A series in powers of
coefficients
centre
e.g.
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Convergence of Power Series
Section 6
Power series often converge for some values of z
but diverge for other values. For example the
series
(geometric series)
converges for ?z ??1 but diverges for ?z ??1
diverges
converges
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Convergence of Power Series
Section 6
Power series often converge for some values of z
but diverge for other values. For example the
series
(geometric series)
converges for ?z ??1 but diverges for ?z ??1
diverges
Radius of Convergence R1
converges
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Example
Radius of Convergence at infinity R??
Section 6
series converges for all z
Example
Zero Radius of Convergence R?0
series diverges for all z (except z?0)
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Section 6
(1) The power series always converges at z?zo
(2) There is a Radius of Convergence R for which
diverges
converges
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Section 6
Is there a quick way to find the radius of
convergence ?
the Cauchy-Hadamard formula
Example
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Section 6
13
Section 6
converges
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Section 6
converges
diverges
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Another Example
Section 6
diverges
converges
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Another Example
Section 6
diverges
converges
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Question
Section 6
Where is the centre of the series? What is the
radius of convergence?
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Note
Section 6
We have used the Cauchy-Hadamard formula to find
the radius of convergence.
There are many other tests (which can be used
when the above test fails). e.g. (1) Root test
(2) if (3) Comparison test (4) The
Ratio test
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Power Series and Analytic functions
Section 6
Every analytic function f (z) can be represented
by a power series with a radius of convergence
R?0. The function is analytic at every point
within the radius of convergence.
Example
series converges for ?z ??1
Radius of Convergence
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Section 6
How do we derive these Power Series?
These power series which represent analytic
functions f (z) are called Taylors series.
They are given by the formula
(Cauchy, 1831)
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Section 6
Example
Derive the Taylor series for
(1) centre z?0
singular point
centre
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Section 6
Example
Derive the Taylor series for
(1) centre z?0
singular point
centre
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Section 6
Example
Derive the Taylor series for
(1) centre z?0
singular point
centre
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Section 6
Example
Derive the Taylor series for
(1) centre z?0
singular point
centre
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Section 6
(2) centre z?1?2
singular point
centre
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An analytic function f (z) can be represented
by different power series with different centres
zo (although there will only be one unique series
for each centre) At least one singular
point of f (z) will be on the circle of
convergence
Section 6
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An analytic function f (z) can be represented
by different power series with different centres
zo (although there will only be one unique series
for each centre) At least one singular
point will be on the circle of convergence
Section 6
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Section 6
Another Example
with centre z?0
no singular points!
centre
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Section 6
Deriving Taylor series directly from the
formula can be very tricky
We usually use other methods (1) Use the
Geometric Series (2) Use the Binomial
Series (3) Use other series (exp., cos,
etc.) (4) Use other tricks
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Section 6
Example
Expand
about z?0
(use the geometric series)
First, draw the centre and singular points to see
whats going on
singular points
So it looks like the radius of convergence should
be R?1
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Section 6
Example
Expand
about z?0
(use the geometric series)
First, draw the centre and singular points to see
whats going on
singular points
centre
So it looks like the radius of convergence should
be R?1
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Section 6
Example
Expand
about z?0
(use the geometric series)
First, draw the centre and singular points to see
whats going on
singular points
centre
So it looks like the radius of convergence should
be R?1
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Section 6
Example
Expand
about z?0
(use the geometric series)
First, draw the centre and singular points to see
whats going on
singular points
centre
So it looks like the radius of convergence should
be R?1
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Section 6
We know that
Therefore
The geometric series converges for ?z??1
Therefore our series converges for ??z2??1
- which is the same as ?z??1, as predicted
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Section 6
We know that
Therefore
The geometric series converges for ?z??1
Therefore our series converges for ??z2??1
- which is the same as ?z??1, as predicted
36
Section 6
We know that
Therefore
The geometric series converges for ?z??1
Therefore our series converges for ??z2??1
- which is the same as ?z??1, as predicted
37
Section 6
We know that
Therefore
The geometric series converges for ?z??1
Therefore our series converges for ??z2??1
- which is the same as ?z??1, as predicted
38
Section 6
Example
Expand
about z?1
(use the geometric series)
First, draw the centre and singular points to see
whats going on
centre
singular point
So it looks like the radius of convergence should
be R?1/2
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Section 6
We know that
Therefore
The geometric series converges for ?z??1
Therefore our series converges for ?2(z?1)??1
- which is the same as ?z ?1 ??1/2, as predicted
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Section 6
Example
Expand
about z?0
(use the binomial series)
First, draw the centre and singular points to see
whats going on
singular point
centre
So it looks like the radius of convergence should
be R?1
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The binomial series is
Section 6
Therefore
as we could guess, since
is singular at z??1
The binomial series converges for ?z??1
Therefore our series converges for ??z??1
- which is the same as ?z??1, as predicted
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The binomial series is
Section 6
Therefore
as we could guess, since
is singular at z??1
The binomial series converges for ?z??1
Therefore our series converges for ??z??1
- which is the same as ?z??1, as predicted
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The binomial series is
Section 6
Therefore
as we could guess, since
is singular at z??1
The binomial series converges for ?z??1
Therefore our series converges for ??z??1
- which is the same as ?z??1, as predicted
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Example
Section 6
Expand
about z?0
singular points
centre
the radius of convergence should be R?2
need to expand in powers of z
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Section 6
Use partial fractions
Now
converges for
46
Section 6
and
converges for
so
converges for
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Section 6
centre
converges inside here
converges inside here
whole thing converges inside overlap
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Section 6
Other useful series
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Example
(of using other series)
Section 6
Expand
about z?0
no singular points the radius of convergence
should be R??
Use the series
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Section 6
Summary
You should be able to find the power series of
a function and its radius of convergence using
one of these methods 1. Taylors
Theorem/Formula 2. Binomial Series 3. Geometric
Series 4. Using other known series, e.g. sin,
cos, exp, etc.
51
Section 6
Summary
You should be able to find the power series of
a function and its radius of convergence using
one of these methods 1. Taylors
Theorem/Formula 2. Binomial Series 3. Geometric
Series 4. Using other known series, e.g. sin,
cos, exp, etc.
52
Section 6
Summary
You should be able to find the power series of
a function and its radius of convergence using
one of these methods 1. Taylors
Theorem/Formula 2. Geometric Series 3.
Geometric Series 4. Using other known series,
e.g. sin, cos, exp, etc.
53
Section 6
Summary
You should be able to find the power series of
a function and its radius of convergence using
one of these methods 1. Taylors
Theorem/Formula 2. Geometric Series 3. Binomial
Series 4. Using other known series, e.g. sin,
cos, exp, etc.
54
Section 6
Summary
You should be able to find the power series of
a function and its radius of convergence using
one of these methods 1. Taylors
Theorem/Formula 2. Geometric Series 3. Binomial
Series 4. Using other known series, e.g. sin,
cos, exp, etc.
55
Section 6
Topics not Covered
(1) Proof of Cauchy-Hadamard formula (follows
from ratio test)
(2) Proof of Taylors Theorem - an analytic
function can be written as a power series (use
Cauchys Integral Formula)
(3) The concept of uniform convergence and proof
that power series are uniformly convergent
(4) Some other practical methods of deriving
power series e.g. use of differentiation/integra
tion of series, differential equations,
undetermined coefficients,
(5) Analytic Continuation
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Section 6
(6) In some very exceptional cases, a singular
point may also arise inside the circle of
convergence.
centre
singular (not analytic) along here
jumps from ?? to ? as we cross this line