ENGG2013 Unit 21 Power Series

1. ENGG2013 Unit 21Power Series Apr, 2011.

2. Charles Kao • Vice-chancellor of CUHK from 1987 to 1996. • Nobel prize laureate in 2009. 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.” kshum

3. 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

4. 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

5. 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

6. Examples • The 0-th order Bessel function of the first kind • The first order Bessel function of the first kind kshum

7. INFINITE SERIES kshum

8. 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+… = 1 diverges diverges diverges kshum

9. Formal definition for convergence • Consider an infinite series • The numbers aimay 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

10. Geometric pictures Complex infinite series Real infinite series Im Complex plane S1 S2 S0 L L- L+ L  Re kshum

11. Convergence of geometric series • If |r|<1, then converges, and the limit is equal to . kshum

12. 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

13. Harmonic series is divergent kshum

14. But is convergent kshum

15. 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

16. Examples is conditionally convergent. is absolutely convergent. kshum

17. 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) • If it is absolutely convergent, then it converges. • (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. http://en.wikipedia.org/wiki/Comparison_test kshum

18. Convergence tests • (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 http://en.wikipedia.org/wiki/Ratio_test kshum

19. Convergence tests • (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. http://en.wikipedia.org/wiki/Root_test kshum

20. 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

21. 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

22. 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

23. Variations: The limit ratio test • If an infinite series z1 + z2 + z3 + … , with all terms nonzero, is such that Then • The series converges if < 1. • The series diverges if  > 1. • No conclusion if  = 1. kshum

24. Variations: The limit root test • If an infinite series z1 + z2 + z3 + … , with all terms nonzero, is such that Then • The series converges if < 1. • The series diverges if  > 1. • No conclusion if  = 1. kshum

25. 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

26. Useful facts • Stirling approximation: for all positive integer n, we have J0(x) converges for every x kshum

27. POWER SERIES kshum

28. General form • The input, x, may be real or complex number. • The coefficient of the nth term, an, may be real or complex number. http://en.wikipedia.org/wiki/Power_series kshum

29. Approximation by tangent line 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‘) kshum

30. Approximation by quadratic 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‘) kshum

31. Third-order 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') kshum

32. Fourth-order 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‘) kshum

33. Fifth-order 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‘) kshum

34. 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

35. Taylor series and Maclaurin series Brook Taylor English mathematician 1685—1731 Colin Maclaurin Scottish mathematician 1698—1746 kshum

36. Examples Geometric series Exponential function Sine function Cosine function More examples at http://en.wikipedia.org/wiki/Maclaurin_series kshum

37. 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

38. 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

39. 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

40. Radius of convergence for GS complex plane 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

41. 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. Im converge Re x0 Proof by comparison test kshum

42. 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. Im converge R c Re x0 Proof by comparison test also kshum

43. 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. Im diverge converge R c Re kshum

44. 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

45. 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. R kshum

46. 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