taylor’s series If f(z) is a complex function analytic inside & on a simple closed curve C (usually a circle) in the z-plane, then the higher derivatives of f(z) also exist inside C. If z0 & z0+h are two fixed points inside C then…..F(z0+h) = f(z0) + hf (1) (z0) + h2/2! f (2) (z0) +…..+ hn /n! f (n) (z0)+….Where f(k)(z0) is the kth derivative of f(z) at z=z0
taylor’s seriesPutting z0+h=z, the series becomes….F(z) = f(z0) + (z-z0) f (1) (z0) + (z-z0)2/2! f (2) (z0) + …..+ (z-z0)n /n! f n (z0) +……. = f (n) (z0).This series is known as Taylor’s series expansion of f(z) about z=z0.The radius of converges of this series is | z – z0 | < R, a disk centered on z=z0 & of radius R.
taylor’s seriesWhen z0=0, in previous equation then we get,F(z) = f(0) + zf (1) (0) + z2/2! f (2) (0) +…..+ zn/n! f (n) (0) +….. The obtained series is Maclaurin’s Series expansion, In the case of function of real variables.
taylor’s series Some Standard power series expansion :-1. ;|z|<∞2. ;|z|<∞3. ;|z|<∞
taylor’s series7. ;|z|<1 ;|z|<18. ( Binomial series for any positive integer m ) ;|z|<1
taylor’s seriesIllustration 5.2: Find the Taylor’s series expansion of f(z)=a/(bz+c) about z=z0. Also determine about the region of convergence.Illustration 5.3: Determine the Taylor’s series expansion of the function f(z)=1/z( z - 2i ) above the point z=i. (a)directly upto the term (z – i)4 , (b) using the binomial expansion. Also determine about the radius of convergence.
taylor’s seriesIllustration 5.4: Find the Taylor’s series expansion of f(z) = 1/( z2 - z – 6 ) about z=1.Illustration 5.5: Expand f(z) = (z - 1)/(z + 1) as Taylor’s series (a) about the point z=0 , and (b) about the point z=1. determine also the result the radius of convergence.
taylor’s seriesIllustration 5.6: (Geometric series) Expand f(z) = 1/(1 – z).Illustration 5.7: Find the Maclaurin’s series of ln[(1 + z)/(1 – z)].Illustration 5.8: Find the Maclaurin’s series of f(z) = 1/(1 – z2).
taylor’s seriesIllustration 5.9: Find the Maclaurin’s series of f(Z) = tan-1 z.Illustration 5.10: (Development by using the geometric series) Develop 1/(c – z) in powers of z – z0, where c – z0 ≠ 0.Illustration 5.11: (Reduction by partial fractions) Find the Taylor’s series of f(z)= with center z0 = 1.
taylor’s seriesIllustration 5.12: Develop f(z) = sin2 z in a Maclaurin’s series and find the radius of convergence.Note: This example can also be solved using the formula sin2 z = ( 1 – cos2z) / 2 & then using the standard power series expansion for cos z with z replaced by 2z.
taylor’s series In the above discussion of power series and in particular Taylor’s series with illustrations, we have seen that inside the radius of convergence, the given function and its Taylor’s series expansion are identically equal. Now the points at which a function fails to be analytic are called Singularities. No Taylor’s series exp. Is possible about singularity. So, Taylor’s exp. About a point z0, at which a function is analytic is only valid within a circle centered z0. Thus all the singularities must be excluded in Taylor’s Expansion.