Power series
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The document discusses convergence of sequences and power series. It defines convergence of a sequence and states that the limit of a convergent sequence is unique. It also discusses Taylor series and Laurent series, stating that if a function f(z) is analytic inside a circle C with center z0, its Taylor series representation about z0 will converge to f(z) for all z inside C. Similarly, if f(z) is analytic in an annular region bounded by two concentric circles, its Laurent series will represent f(z) in that region.
Power series
- 1. Power Series N. B. Vyas Department of Mathematics, Atmiya Institute of Tech. and Science, Rajkot (Guj.) N.B.V yas − Department of M athematics, AIT S − Rajkot
- 2. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that N.B.V yas − Department of M athematics, AIT S − Rajkot
- 3. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N N.B.V yas − Department of M athematics, AIT S − Rajkot
- 4. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 5. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ A sequence which is not convergent is defined to be divergent. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 6. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ A sequence which is not convergent is defined to be divergent. If lim zn = z0 we have n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 7. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ A sequence which is not convergent is defined to be divergent. If lim zn = z0 we have n→∞ (i) |zn | → |z0 | as n → ∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 8. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ A sequence which is not convergent is defined to be divergent. If lim zn = z0 we have n→∞ (i) |zn | → |z0 | as n → ∞ (ii) the sequence {zn } is bounded N.B.V yas − Department of M athematics, AIT S − Rajkot
- 9. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ A sequence which is not convergent is defined to be divergent. If lim zn = z0 we have n→∞ (i) |zn | → |z0 | as n → ∞ (ii) the sequence {zn } is bounded If zn = xn + iyn and z0 = x0 + iy0 then N.B.V yas − Department of M athematics, AIT S − Rajkot
- 10. Convergence of a Sequence A sequence {zn } is said to be converge to z0 {as n approaches inf inity} if, for each > 0 there exists a positive integer N such that |z − z0 | < , whenever n≥N Symbolically we write lim zn = z0 n→∞ A sequence which is not convergent is defined to be divergent. If lim zn = z0 we have n→∞ (i) |zn | → |z0 | as n → ∞ (ii) the sequence {zn } is bounded If zn = xn + iyn and z0 = x0 + iy0 then lim zn = z0 ⇒ lim xn = x0 and lim yn = y0 n→∞ n→∞ n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 11. Convergence of a Sequence The limit of convergent sequence is unique. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 12. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 13. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 14. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ 2 lim czn = cz n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 15. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ 2 lim czn = cz n→∞ 3 lim zn wn = zw n→∞ N.B.V yas − Department of M athematics, AIT S − Rajkot
- 16. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ 2 lim czn = cz n→∞ 3 lim zn wn = zw n→∞ zn z 4 lim = (w = 0) n→∞ wn w N.B.V yas − Department of M athematics, AIT S − Rajkot
- 17. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ 2 lim czn = cz n→∞ 3 lim zn wn = zw n→∞ zn z 4 lim = (w = 0) n→∞ wn w Given a sequence {an }. Consider a sequence {nk } of positive integers such that n1 < n2 < n3 < . . . then the sequence {ank } is called a subsequence of {an }. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 18. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ 2 lim czn = cz n→∞ 3 lim zn wn = zw n→∞ zn z 4 lim = (w = 0) n→∞ wn w Given a sequence {an }. Consider a sequence {nk } of positive integers such that n1 < n2 < n3 < . . . then the sequence {ank } is called a subsequence of {an }. If {ank } converges then its limit is called Sub-sequential limit N.B.V yas − Department of M athematics, AIT S − Rajkot
- 19. Convergence of a Sequence The limit of convergent sequence is unique. lim zn = z and lim wn = w then n→∞ n→∞ 1 lim (zn ± wn ) = z + w n→∞ 2 lim czn = cz n→∞ 3 lim zn wn = zw n→∞ zn z 4 lim = (w = 0) n→∞ wn w Given a sequence {an }. Consider a sequence {nk } of positive integers such that n1 < n2 < n3 < . . . then the sequence {ank } is called a subsequence of {an }. If {ank } converges then its limit is called Sub-sequential limit A sequence {an } of complex numbers converges to p if and only if every subsequence converges to p. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 20. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C N.B.V yas − Department of M athematics, AIT S − Rajkot
- 21. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C (z − z0 )2 (z − z0 )n n f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 ) 2! n! N.B.V yas − Department of M athematics, AIT S − Rajkot
- 22. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C (z − z0 )2 (z − z0 )n n f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 ) 2! n! Case 1: Putting z = z0 + h in above equation, we get N.B.V yas − Department of M athematics, AIT S − Rajkot
- 23. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C (z − z0 )2 (z − z0 )n n f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 ) 2! n! Case 1: Putting z = z0 + h in above equation, we get h2 hn f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 ) 2! n! N.B.V yas − Department of M athematics, AIT S − Rajkot
- 24. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C (z − z0 )2 (z − z0 )n n f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 ) 2! n! Case 1: Putting z = z0 + h in above equation, we get h2 hn f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 ) 2! n! Case 2: If z0 = 0 then, we get N.B.V yas − Department of M athematics, AIT S − Rajkot
- 25. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C (z − z0 )2 (z − z0 )n n f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 ) 2! n! Case 1: Putting z = z0 + h in above equation, we get h2 hn f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 ) 2! n! Case 2: If z0 = 0 then, we get z2 zn f (z) = f (0) + zf (0) + f (0) + . . . + f n (0) 2! n! N.B.V yas − Department of M athematics, AIT S − Rajkot
- 26. Taylors Series If f (z) is analytic inside a circle C with centre at z0 then for all z inside C (z − z0 )2 (z − z0 )n n f (z) = f (z0 )+(z−z0 )f (z0 )+ f (z0 )+. . .+ f (z0 ) 2! n! Case 1: Putting z = z0 + h in above equation, we get h2 hn f (z0 + h) = f (z0 ) + hf (z0 ) + f (z0 ) + . . . + f n (z0 ) 2! n! Case 2: If z0 = 0 then, we get z2 zn f (z) = f (0) + zf (0) + f (0) + . . . + f n (0) 2! n! This series is called Maclaurin’s Series. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 27. Laurent Series If f (z) is analytic in the ring shaped region R bounded by two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and with centre at z0 , then for all z in R. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 28. Laurent Series If f (z) is analytic in the ring shaped region R bounded by two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and with centre at z0 , then for all z in R. b1 b2 f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . . (z − z0 ) (z − z0 )2 N.B.V yas − Department of M athematics, AIT S − Rajkot
- 29. Laurent Series If f (z) is analytic in the ring shaped region R bounded by two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and with centre at z0 , then for all z in R. b1 b2 f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . . (z − z0 ) (z − z0 )2 1 f (ξ)dξ where an = , n = 0, 1, 2, . . . 2πi Γ (ξ − z0 )n+1 N.B.V yas − Department of M athematics, AIT S − Rajkot
- 30. Laurent Series If f (z) is analytic in the ring shaped region R bounded by two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and with centre at z0 , then for all z in R. b1 b2 f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . . (z − z0 ) (z − z0 )2 1 f (ξ)dξ where an = , n = 0, 1, 2, . . . 2πi Γ (ξ − z0 )n+1 Γ being any circle lying between c1 & c2 having z0 as its centre, for all values of n. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 31. Laurent Series If f (z) is analytic in the ring shaped region R bounded by two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and with centre at z0 , then for all z in R. b1 b2 f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . . (z − z0 ) (z − z0 )2 1 f (ξ)dξ where an = , n = 0, 1, 2, . . . 2πi Γ (ξ − z0 )n+1 Γ being any circle lying between c1 & c2 having z0 as its centre, for all values of n. ∞ ∞ n bn ∴ f (z) = an (z − z0 ) + n=0 n=1 (z − z0 )n N.B.V yas − Department of M athematics, AIT S − Rajkot
- 32. Laurent Series If f (z) is analytic in the ring shaped region R bounded by two concentric circles c1 & c2 of radii r1 & r2 (r1 > r2 ) and with centre at z0 , then for all z in R. b1 b2 f (z) = a0 +a1 (z −z0 )+a2 (z −z0 )2 +. . .+ + +. . . (z − z0 ) (z − z0 )2 1 f (ξ)dξ where an = , n = 0, 1, 2, . . . 2πi Γ (ξ − z0 )n+1 Γ being any circle lying between c1 & c2 having z0 as its centre, for all values of n. ∞ ∞ n bn ∴ f (z) = an (z − z0 ) + n=0 n=1 (z − z0 )n N.B.V yas − Department of M athematics, AIT S − Rajkot
- 33. Note If f (z) is analytic at z = z0 then we can expand f (z) by means of Taylor’s series at a point z0 N.B.V yas − Department of M athematics, AIT S − Rajkot
- 34. Note If f (z) is analytic at z = z0 then we can expand f (z) by means of Taylor’s series at a point z0 Laurent series given an expansion of f (z) at a point z0 even if f (z) is not analytic there. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 35. Singular Points A point at which a function f (z) ceases to be analytic is called a singular point of f (z) N.B.V yas − Department of M athematics, AIT S − Rajkot
- 36. Singular Points A point at which a function f (z) ceases to be analytic is called a singular point of f (z) If the function f (z) is analytic at every point in the neighbourhood of a point z0 except at z0 is called isolated singular point or isolated singularity. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 37. Singular Points A point at which a function f (z) ceases to be analytic is called a singular point of f (z) If the function f (z) is analytic at every point in the neighbourhood of a point z0 except at z0 is called isolated singular point or isolated singularity. 1 1 Eg. 1 f (z) = ⇒ f (z) = − 2 , it follows that f (z) is analytic at z z every point except at z = 0 , hence z = 0 is an isolated singularity. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 38. Singular Points A point at which a function f (z) ceases to be analytic is called a singular point of f (z) If the function f (z) is analytic at every point in the neighbourhood of a point z0 except at z0 is called isolated singular point or isolated singularity. 1 1 Eg. 1 f (z) = ⇒ f (z) = − 2 , it follows that f (z) is analytic at z z every point except at z = 0 , hence z = 0 is an isolated singularity. 1 Eg. 2 f (z) = 3 2 has three isolated singularities at z (z + 1) z = 0, i, −i N.B.V yas − Department of M athematics, AIT S − Rajkot
- 39. Singular Points If z = z0 is a isolated singular point, then f (z) can be expanded in a Laurents series in the form. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 40. Singular Points If z = z0 is a isolated singular point, then f (z) can be expanded in a Laurents series in the form. ∞ ∞ n bn f (z) = an (z − z0 ) + n=0 n=1 (z − z0 )n N.B.V yas − Department of M athematics, AIT S − Rajkot
- 41. Singular Points If z = z0 is a isolated singular point, then f (z) can be expanded in a Laurents series in the form. ∞ ∞ n bn f (z) = an (z − z0 ) + (1) n=0 n=1 (z − z0 )n ∞ In (1) an (z − z0 )n is called the regular part and n=0 ∞ bn is called the principal part of f (z) in the n=1 (z − z0 )n neighbourhood of z0 . N.B.V yas − Department of M athematics, AIT S − Rajkot
- 42. Singular Points If z = z0 is a isolated singular point, then f (z) can be expanded in a Laurents series in the form. ∞ ∞ n bn f (z) = an (z − z0 ) + (1) n=0 n=1 (z − z0 )n ∞ In (1) an (z − z0 )n is called the regular part and n=0 ∞ bn is called the principal part of f (z) in the n=1 (z − z0 )n neighbourhood of z0 . If the principal part of f (z) contains infinite numbers of terms then z = z0 is called an isolated essential singularity of f (z). N.B.V yas − Department of M athematics, AIT S − Rajkot
- 43. Singular Points If z = z0 is a isolated singular point, then f (z) can be expanded in a Laurents series in the form. ∞ ∞ n bn f (z) = an (z − z0 ) + (1) n=0 n=1 (z − z0 )n ∞ In (1) an (z − z0 )n is called the regular part and n=0 ∞ bn is called the principal part of f (z) in the n=1 (z − z0 )n neighbourhood of z0 . If the principal part of f (z) contains infinite numbers of terms then z = z0 is called an isolated essential singularity of f (z). N.B.V yas − Department of M athematics, AIT S − Rajkot
- 44. Singular Points If in equation(1) , the principal part has all the coefficient bn+1 , bn+2 , . . . as zero after a particular term bn then the Laurents series of f (z) reduces to N.B.V yas − Department of M athematics, AIT S − Rajkot
- 45. Singular Points If in equation(1) , the principal part has all the coefficient bn+1 , bn+2 , . . . as zero after a particular term bn then the Laurents series of f (z) reduces to ∞ b1 b2 bn f (z) = an (z − z0 )n + + 2 + ... + n=0 (z − z0 ) (z − z0 ) (z − z0 )n N.B.V yas − Department of M athematics, AIT S − Rajkot
- 46. Singular Points If in equation(1) , the principal part has all the coefficient bn+1 , bn+2 , . . . as zero after a particular term bn then the Laurents series of f (z) reduces to ∞ b1 b2 bn f (z) = an (z − z0 )n + + 2 + ... + n=0 (z − z0 ) (z − z0 ) (z − z0 )n i.e. (Regular part) + (Principal part is a polynomial of finite 1 number of terms in z − z0 N.B.V yas − Department of M athematics, AIT S − Rajkot
- 47. Singular Points If in equation(1) , the principal part has all the coefficient bn+1 , bn+2 , . . . as zero after a particular term bn then the Laurents series of f (z) reduces to ∞ b1 b2 bn f (z) = an (z − z0 )n + + 2 + ... + n=0 (z − z0 ) (z − z0 ) (z − z0 )n i.e. (Regular part) + (Principal part is a polynomial of finite 1 number of terms in z − z0 The the singularity in this case at z = z0 is called a pole of order n. N.B.V yas − Department of M athematics, AIT S − Rajkot
- 48. Singular Points If in equation(1) , the principal part has all the coefficient bn+1 , bn+2 , . . . as zero after a particular term bn then the Laurents series of f (z) reduces to ∞ b1 b2 bn f (z) = an (z − z0 )n + + 2 + ... + n=0 (z − z0 ) (z − z0 ) (z − z0 )n i.e. (Regular part) + (Principal part is a polynomial of finite 1 number of terms in z − z0 The the singularity in this case at z = z0 is called a pole of order n. If the order of the pole is one, the pole is called simple pole. N.B.V yas − Department of M athematics, AIT S − Rajkot