ENGG2013 Unit 21

Power Series

Apr, 2011.

Charles Kao

• Vice-chancellor of CUHK from 1987 to 1996.

• Nobel prize laureate in 2009.

kshum 2

K. C. Kao and G. A. Hockham, "Dielectric-fibre surface waveguides for optical frequencies," Proc. IEE, vol. 133, no. 7, pp.1151–1158, 1966.

“It is foreseeable that glasses with a bulk loss of about 20 dB/km at around 0.6 micrometer will be obtained, as the iron impurity concentration may be reduced to 1 part per million.”

Special functions

From the first paragraph of Prof. Kao’s paper (after abstract), we see

• Jn = nth-order Bessel function of the first kind

• Kn = nth-order modified Bessel function of the second kind.

• H(i)= th-order Hankel function of the ith

type.kshum 3

J(x)• There is a parameter called the “order”.• The th-order Bessel function of the first kind

– http://en.wikipedia.org/wiki/Bessel_function

• Two different definitions:– Defined as the solution to the differential

equation

– Defined by power series:

kshum 4

Gamma function (x)

• Gamma function is the extension of the factorial function to real integer input.– http://en.wikipedia.org/wiki/Gamma_function

• Definition by integral

• Property : (1) = 1, and for integer n, (n)=(n – 1)!

kshum 5

Examples

• The 0-th order Bessel function of the first kind

• The first order Bessel function of the first kind

kshum 6

INFINITE SERIES

kshum 7

Infinite series• Geometric series

– If a = 1 and r= 1/2,

– If a = 1 and r = 1 1+1+1+1+1+…– If a = 1 and r = – 1

1 – 1 + 1 – 1 + 1 – 1 + …– If a = 1 and r = 2 1+2+4+8+16+…

kshum 8

= 1

diverges

diverges

diverges

Formal definition for convergence• Consider an infinite series

– The numbers ai may be real or complex.

• Let Sn be the nth partial sum

• The infinite series is said to be convergent if there is a number L such that, for every arbitrarily small > 0, there exists an integer N such that

• The number L is called the limit of the infinite series.kshum 9

Geometric pictures

kshum 10

Complex infinite series

Complex plane

Re

Im

L

Real infinite series

L L+L-

S0

S1S2

Convergence of geometric series

• If |r|<1, then converges, and the limit

is equal to .

kshum 11

Easy fact

• If the magnitudes of the terms in an infinite series does not approach zero, then the infinite series diverges.

• But the converse is not true.

kshum 12

Harmonic series

kshum 13

is divergent

But

kshum 14

is convergent

Terminologies

• An infinite series z1+z2+z3+… is called absolutely convergent if |z1|+|z2|+|z3|+… is convergent.

• An infinite series z1+z2+z3+… is called conditionally convergent if z1+z2+z3+… is convergent, but |z1|+|z2|+|z3|+… is divergent.

kshum 15

Examples

•

is conditionally convergent.

•

is absolutely convergent.

kshum 16

Convergence tests

Some sufficient conditions for convergence.Let z1 + z2 + z3 + z4 + … be a given infinite series.(z1, z2, z3, … are real or complex numbers)1. If it is absolutely convergent, then it converges.2. (Comparison test) If we can find a convergent

series b1 + b2 + b3 + … with non-negative real terms such that

|zi| bi for all i, then z1 + z2 + z3 + z4 + … converges.

kshum 17

http://en.wikipedia.org/wiki/Comparison_test

Convergence tests

3. (Ratio test) If there is a real number q < 1, such that

for all i > N (N is some integer), then z1 + z2 + z3 + z4 + … converges.

If for all i > N , , then it diverges

kshum 18

http://en.wikipedia.org/wiki/Ratio_test

Convergence tests

4. (Root test) If there is a real number q < 1, such that

for all i > N (N is some integer),then z1 + z2 + z3 + z4 + … converges.

If for all i > N , , then it diverges.

kshum 19

http://en.wikipedia.org/wiki/Root_test

Derivation of the root test from comparison test

• Suppose that for all i N. Then

for all i N. But

is a convergent series (because q<1). Therefore z1 + z2 + z3 + z4 + … converges by the comparison test.

kshum 20

Application

• Given a complex number x, apply the ratio test to

• The ratio of the (i+1)-st term and the i-th term is

Let q be a real number strictly less than 1, say q=0.99. Then,

Therefore exp(x) is convergent for all complex number x.

kshum 21

Application

• Given a complex number x, apply the root test to

• The ratio of the (i+1)-st term and the i-th term is

Let q be a real number strictly less than 1, say q=0.99. Then,

Therefore exp(x) is convergent for all complex number x.

kshum 22

Variations: The limit ratio test

• If an infinite series z1 + z2 + z3 + … , with all terms nonzero, is such that

Then1.The series converges if < 1.2.The series diverges if > 1.3.No conclusion if = 1.

kshum 23

Variations: The limit root test

• If an infinite series z1 + z2 + z3 + … , with all terms nonzero, is such that

Then1.The series converges if < 1.2.The series diverges if > 1.3.No conclusion if = 1.

kshum 24

Application

• Let x be a given complex number. Apply the limit root test to

• The nth term is

• The nth root of the magnitude of the nth term is

kshum 25

Useful facts

• Stirling approximation: for all positive integer n, we have

•

kshum 26

J0(x) converges for every x

POWER SERIES

kshum 27

General form

• The input, x, may be real or complex number.• The coefficient of the nth term, an, may be real

or complex number.

kshum 28

http://en.wikipedia.org/wiki/Power_series

Approximation by tangent line

kshum 29

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

y = log(x)

Tangent line at x=0.6

x = linspace(0.1,2,50);plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6,'b')grid on; xlabel('x'); ylabel('y');legend(‘y = log(x)’, ‘Tangent line at x=0.6‘)

Approximation by quadratic

kshum 30

x = linspace(0.1,2,50);plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2,'b')grid on; xlabel('x'); ylabel('y')legend(‘y = log(x)’, ‘Second-order approx at x=0.6‘)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1.5

-1

-0.5

0

0.5

1

x

y

y = log(x)

Second-order approx at x=0.6

Third-order

kshum 31

x = linspace(0.05,2,50);plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2+(x-0.6).^3/0.6^3/3,'b')grid on; xlabel('x'); ylabel('y')legend('y = log(x)', ‘Third-order approx at x=0.6')

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3

4

x

y

y = log(x)

Third-order approx at x=0.6

Fourth-order

kshum 32

x = linspace(0.05,2,50);plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2+(x-0.6).^3/0.6^3/3-(x-0.6).^4/0.6^4/4,'b')grid on; xlabel('x'); ylabel('y')legend(‘y = log(x)’, ‘Fourth-order approx at x=0.6‘)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5

-4

-3

-2

-1

0

1

x

y

y = log(x)

Fourth-order approx at x=0.6

Fifth-order

kshum 33

x = linspace(0.05,2,50);plot(x,log(x),'r',x, log(0.6)+(x-0.6)/0.6-(x-0.6).^2/0.6^2/2+(x-0.6).^3/0.6^3/3-(x-0.6).^4/0.6^4/4+(x-0.6).^5/0.6^5/5,'b')grid on; xlabel('x'); ylabel('y')legend(‘y = log(x)’, ‘Fifth-order approx at x=0.6‘)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-2

0

2

4

6

8

10

x

y

y = log(x)

Fifth-order approx at x=0.6

Taylor series• Local approximation by power series.

• Try to approximate a function f(x) near x0, by

a0 + a1(x – x0) + a2(x – x0)2 + a3(x – x0)3 + a4(x – x0)4 + …

• x0 is called the centre.• When x0 = 0, it is called Maclaurin series.

a0 + a1x + a2 x2 + a3 x3 + a4x4 + a5x5 + a6x6 + …

kshum 34

Taylor series and Maclaurin series

kshum 35

Brook TaylorEnglish mathematician1685—1731

Colin MaclaurinScottish mathematician1698—1746

Examples

kshum 36

Geometric series

Exponential function

Sine function

Cosine function

More examples at http://en.wikipedia.org/wiki/Maclaurin_series

How to obtain the coefficients• Match the derivatives at x =x0

• Set x = x0 in f(x) = a0+a1(x – x0)+a2(x – x0)2 +a3(x – x0)3+...

a0= f(x0)

• Set x = x0 in f’(x) = a1+2a2(x – x0) +3a3(x – x0)2+… a1= f’(x0)

• Set x = x0 in f’’(x) = 2a2+6a3(x – x0)+12a4(x – x0)2+…

a2= f’’(x0)/2– In general, we have ak= f(k)(x0) / k!

kshum 37

Example f(x) = log(x), x0=0.6• First-order approx.

log(0.6)+(x – 0.6)/0.6• Second-order approx.

log(0.6)+(x – 0.6)/0.6 – (x – 0.6)2/(2· 0.62)• Third-order approx.

log(0.6)+(x–0.6)/0.6 – (x–0.6)2/(2· 0.62) +(x–0.6)3/(3· 0.63)

kshum 38

Example: Geometric series• Maclaurin series 1/(1– x) = 1+x+x2+x3+x4+x5+x6+…• Equality holds when |x| < 1

• If we carelessly substitute x=1.1, then L.H.S. of 1/(1– x) = 1+x+x2+x3+x4+x5+x6+…is equal to -10, but R.H.S. is not well-defined.

kshum 39

Radius of convergence for GS• For the geometric series 1+z+z2+z3+… , it

converges if |z| < 1, but diverges when |z| > 1.

• We say that the radius of convergence is 1.• 1+z+z2+z3+… converges inside the unit disc,

and diverges outside.

kshum 40

complex plane

Convergence of Maclaurin series in general

• If the power series f(x) converges at a point x0, then it converges for all x such that |x| < |x0| in the complex plane.

kshum 41

x0

Re

Im

conve

rge

Proof by comparison test

Convergence of Taylor series in general

• If the power series f(x) converges at a point x0, then it converges for all x such that

|x – c| < |x0 – c| in the complex plane.

kshum 42

x0

Re

Im

conve

rge

Proof by comparison test also

cR

Region of convergence

• The region of convergence of a Taylor series with center c is the smallest circle with center c, which contains all the points at which f(x) converges.

• The radius of the region of convergence is called the radius of convergence of this Taylor series.

kshum 43

Re

Im

conve

rge

cR

diverge

Examples

• : radius of convergence = 1. It converges

at the point z= –1, but diverges for all |z|>1. • exp(z): radius of convergence is , because it

converges everywhere.• : radius of convergence is 0, because

it diverges everywhere except z=0. kshum 44

Behavior on the circle of convergence

• On the circle of convergence |z-c| = R, a Taylor series may or may not converges.

• All three series zn, zn/n, and zn/n2

Have the same radius of convergence R=1.

But zn diverges everywhere on |z|=1, zn /n diverges at z= 1 and converges at z=– 1 , zn/n2 converges everywhere on |z|=1.

kshum 45

R

Summary

• Power series is useful in calculating special functions, such as exp(x), sin(x), cos(x), Bessel functions, etc.

• The evaluation of Taylor series is limited to the points inside a circle called the region of convergence.

• We can determine the radius of convergence by root test, ratio test, etc.

kshum 46