NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
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1 NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 17
2 Complex Analysis Module: 5: Consequences of Complex Integration Lecture: 2: Taylor series A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 2 / 17
3 Consequences of complex integration Taylor Series A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 3 / 17
4 In this part, we find certain series expansions that are consequences of Cauchy Integral Formula. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 4 / 17
5 Taylor theorem Theorem Let f (z) be analytic at all points within a circle C 0 with center z 0 and radius ρ 0. Then for every point z within C 0, we have f (z) = f (z 0 ) + f (z 0 )(z z 0 ) + f (z 0 ) (z z 0 ) f n 2! n! (z z 0) n +... (z z 0 ) n = f (z 0 ) + f (n) (z z 0 ) n (z 0 ) = f (n) (z 0 ), n! n! n=1 where the power series converges to f (z) on the disc z z 0 < ρ 0. n=0 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 5 / 17
6 The representation of f (z) as an infinite series (z z 0 ) n f (z) = f (n) (z 0 ) is known as Taylor Series. n! n=0 Using the Taylor series expansion, we can write f (n) (z 0 ) = 1 f (z) dz. 2πi (z z 0 ) n+1 We prove this result using Cauchy Integral formula. When z 0 = 0, this reduces to f (z) = f (0) + c n=1 which is known as Maclaurin s series. z n n! f (n) (0) Taylor s series at origin is called Maclaurin s series. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 6 / 17
7 For the validity of the expansion as a Taylor series, it is essential that f (z) be analytic at all points inside the circle C 0 for then the convergence of Taylor series for f (z) is assured. Hence the greatest radius of C 0 is the distance from the point z 0 to the singularity of f (z) which is nearest to z 0, since we require the function to be analytic at all points within C 0. If f (z) is analytic in a region Ω, containing the point a such that f (a) and all its derivatives f n (a) vanishes, then f(z)=0 in Ω. If f is analytic at z 0, the Taylor series for f around z 0 can be obtained by termwise differentiation of the Taylor series for f around z 0 and converges in the same disk as the series for f. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 7 / 17
8 Let f and g be analytic functions with Taylor series f (z) = a j (z z 0 ) j and g(z) = b j (z z 0 ) j around the point z 0, then (i) the Taylor series for cf(z), c a constant, is (ii) the Taylor series for f (z) ± g(z) is ca j (z z 0 ) j ; (a j ± b j )(z z 0 ) j. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 8 / 17
9 The Cauchy product of two Taylor series f (z) = and g(z) = h(z) = a j (z z 0 ) j b j (z z 0 ) j is defined to be the series c j (z z 0 ) j, where c j = j a j i b i. Let f and g be analytic functions with Taylor series f (z) = a j (z z 0 ) j and g(z) = b j (z z 0 ) j around the point z 0. Then the Taylor series for the product fg around z 0 is given by the Cauchy product of these two series. i=0 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 9 / 17
10 Comparison with real functions An analytic function is infinitely differentiable and always has a Taylor series expansion with a non-zero radius of convergence. A real function f can also be infinitely differentiable. But it is sometimes impossible to expand it in a power series. An example of such a function is { e 1/x 2, x 0 f (x) = 0, x = 0. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 10 / 17
11 Comparison with real functions If a real valued function f (x) defined on the whole of real line is extended as complex valued function f (z) analytically, it may not be analytic in the entire complex plane. For example, e x is real analytic function for all real x and its complex extension e z is also an analytic (entire) function. Whereas, 1/(1 + x 2 ) is a real analytic function for all real x. But its complex extension 1/(1 + z 2 ) ceases to be analytic at z = ±i A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 11 / 17
12 Example The function f (z) = exp(z) is entire. Hence Macluaurin series expansion is valid for all z. Since f (n) (0) = 1 we get f (z) = exp(z) = n=0 z n n! A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 12 / 17
13 Example The function f (z) = sin z is entire. Hence Macluaurin series expansion is valid for all z. Since f (2n) (0) = 0 and f (2n 1) (0) = ( 1) n, we get f (z) = sin z = ( 1) n z 2n+1 (2n + 1)! n=0 Differentiating both sides of the function gives cos z = ( 1) n z2n (2n)! n=0 Since sinh z = i sin(iz), the Maclaurin series expansion for sinh z and cosh z are also possible. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 13 / 17
14 Example Expand log(1 + z) in a Taylor Series about z=0 and determine the region of convergence for the resulting series. Solution: Here f (z) = log(1 + z). We find f (z), f (z), f (z)... f n (z) and their values at the point 0. Using the above values, we write the series as f (z) = z z2 2 + z3 3 + z4 zn ( 1)n 1 4 n +... The point z = 1 is the singularity of log(1 + z) nearest the point z=0. Thus the series converges for all values of z within the circle z = 1. A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 14 / 17
15 Example Expand about the point z 0 = 2i. f (z) = 1 1 z A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 15 / 17
16 We illustrate the use of geometric series in deriving the Taylor series. Solution Write 1 1 z = 1 1 z + 2i 2i = i 1 z 2i 1 2i = 1 1 2i + 1 (1 2i) 2 (z 2i) + 1 (1 2i) 3 (z 2i)2 + The distance from the center z 0 = 2i to the nearest singularity z = 1, is 5. Hence the circle of convergence is z 2i < 5 A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 16 / 17
17 Exercises Expand in a Maclaurin series and also give the radius of convergence of the function f (z) = ze z2 Use Maclaurin series for e z to expand about z 0 = 1. f (z) = (z 1)e z A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 17 / 17
NPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 19 Complex Analysis Module: 8:
More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 18 Complex Analysis Module: 5:
More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 28 Complex Analysis Module: 6:
More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 14 Complex Analysis Module: 1:
More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
NPTEL web course on omplex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) omplex Analysis 1 / 18 omplex Analysis Module: 6: Residue
More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
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More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
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More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
NPTEL web course on Complex Analysis A. Swaminathan I.I.T. Roorkee, India and V.K. Katiyar I.I.T. Roorkee, India A.Swaminathan and V.K.Katiyar (NPTEL) Complex Analysis 1 / 27 Complex Analysis Module: 2:
More informationNPTEL web course on Complex Analysis. A. Swaminathan I.I.T. Roorkee, India. and. V.K. Katiyar I.I.T. Roorkee, India
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