2. 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
3. taylor’s series
Putting 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.
4. taylor’s series
When 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.
7. taylor’s series
7. ;|z|<1
;|z|<1
8. ( Binomial series for any positive integer m )
;|z|<1
8. taylor’s series
Illustration 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.
9. taylor’s series
Illustration 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.
10. taylor’s series
Illustration 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).
11. taylor’s series
Illustration 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.
12. taylor’s series
Illustration 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.
13. 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.
