engg2013 unit 21 power series apr, 2011.. charles kao vice-chancellor of cuhk from 1987 to 1996....
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ENGG2013 Unit 21
Power Series
Apr, 2011.
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Charles Kao
• Vice-chancellor of CUHK from 1987 to 1996.
• Nobel prize laureate in 2009.
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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.”
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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
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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:
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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)!
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Examples
• The 0-th order Bessel function of the first kind
• The first order Bessel function of the first kind
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INFINITE SERIES
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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+…
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= 1
diverges
diverges
diverges
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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
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Geometric pictures
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Complex infinite series
Complex plane
Re
Im
L
Real infinite series
L L+L-
S0
S1S2
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Convergence of geometric series
• If |r|<1, then converges, and the limit
is equal to .
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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.
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Harmonic series
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is divergent
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But
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is convergent
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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.
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Examples
•
is conditionally convergent.
•
is absolutely convergent.
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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.
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http://en.wikipedia.org/wiki/Comparison_test
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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
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http://en.wikipedia.org/wiki/Ratio_test
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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.
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http://en.wikipedia.org/wiki/Root_test
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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.
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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.
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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.
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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.
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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.
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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
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Useful facts
• Stirling approximation: for all positive integer n, we have
•
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J0(x) converges for every x
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POWER SERIES
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General form
• The input, x, may be real or complex number.• The coefficient of the nth term, an, may be real
or complex number.
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http://en.wikipedia.org/wiki/Power_series
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Approximation by tangent line
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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‘)
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Approximation by quadratic
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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
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Third-order
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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
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Fourth-order
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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
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Fifth-order
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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
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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 + …
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Taylor series and Maclaurin series
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Brook TaylorEnglish mathematician1685—1731
Colin MaclaurinScottish mathematician1698—1746
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Examples
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Geometric series
Exponential function
Sine function
Cosine function
More examples at http://en.wikipedia.org/wiki/Maclaurin_series
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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!
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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)
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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.
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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.
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complex plane
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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.
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x0
Re
Im
conve
rge
Proof by comparison test
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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.
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x0
Re
Im
conve
rge
Proof by comparison test also
cR
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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.
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Re
Im
conve
rge
cR
diverge
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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
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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.
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R
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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.
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