Taylor’s Series - Advanced Engineering Math - Lecture Slides, Slides of Engineering Mathematics

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Taylor’s series

Recall from calculus

• Infinite equence of numbers

• Infinite series

• Power series

1+x+2x

2

+3x

3

+4x

4

Power series for approximation

• Demonstration using Sage

– Sage is a free mathematical software.

http://wiki.sagemath.org/interact/

Charles Kao’s paper on fibre optics

• “Dielectric-fibre surface waveguides for optical

frequencies”, by K. C. Kao and G. A. Hockman,

Proc. IEE, vol. 133, no. 7, pp.1151–1158, 1966.

http://home.deib.polimi.it/martinel/comunicazioni/kaonobelpaper.pdf … glasses with a bulk loss of about 20 dB/km at around 0.6 micrometer can be obtained …

Power series for the calculating

special functions

• Compare with the sine function.

• Sine function sin(x) can be defined as a solution

to the second-order differential equation

where  is a parameter.

• We can calculate it by power series

The gamma function

• The gamma function can be regarded as an

interpolation of the factorial function

• For positive integer n,

(n) = (n-1)!

• It satisfies a recursive formula

(x+1) = x (x)

for positive real number x. We can write it as (x) =

(x+1)/x

• We can calculate the gamma function using the built-in

calculator in iphone

• For all positive real number x

(x) = (x)! / x

Factorial function in the landscape mode of iphone’s calculator

Taylor series

• Given a function f(x), and a point x

0

• x 0 is called the centre.

• Try to approximate a function f(x) near x

0

, by

a

0

+ a

1

(x – x

0

) + a

2

(x – x

0

2

+ a

3

(x – x

0

3

+ a

4

(x – x

0

4

• When x

0

= 0, it is called Maclaurin series.

a

0

+ a

1

x + a

2

x

2

+ a

3

x

3

+ a

4

x

4

+ a

5

x

5

+ a

6

x

6

http://en.wikipedia.org/wiki/Power_series

Taylor series and Maclaurin series

Brook Taylor English mathematician 1685 — 1731 Colin Maclaurin Scottish mathematician 1698 — 1746

Maclaurin series of sine

• Equate sin(x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + …
• Set x = 0  a 0 = sin(0) = 0.
• Differentiate f and set x = 0, cos(x) = a 1 + 2a 2 x + 3a 3 x 2 + 4a 4 x 3 + …  a 1 =cos(0)=1.
• Differentiate f again and set x = 0, -sin(x) = 2a 2 + 6a 3 x + 12a 4 x 2 + …  a 2 =sin(0)/2=0.
• Differentiate f again and set x = 0, -cos(x) = 6a 3 + 24a 4 x + 60a 5 x 2 + …  a 3 =-cos(0)/6=-1/6.
• Differentiate f again and set x = 0, sin(x) = 24a 4 + 120a 5 x + 360a 6 x 2 + …  a 4 =sin(0)/24=0. By mathematical induction, we can obtain all coefficients.

Maclaurin series of cosine

• Equate cos(x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + …
• Set x = 0  a 0 = cos(0) = 1.
• Differentiate f and set x = 0, -sin(x) = a 1 + 2a 2 x + 3a 3 x 2 + 4a 4 x 3 + …  a 1 =-sin(0)=0.
• Differentiate f again and set x = 0, -cos(x) = 2a 2 + 6a 3 x + 12a 4 x 2 + …  a 2 =-cos(0)/2=-1/2.
• Differentiate f again and set x = 0, sin(x) = 6a 3 + 24a 4 x + 60a 5 x 2 + …  a 3 =sin(0)/6=0.
• Differentiate f again and set x = 0, cos(x) = 24a 4 + 120a 5 x + 360a 6 x 2 + …  a 4 =cos(0)/24=1/24. By mathematical induction, we can obtain all coefficients.

Important note

• Let f(x) be a smooth function (so that we can

differentiate it arbitrarily many times).

• Let p(x) be the Taylor series expansion of f(x) with

centre x

0

• It is not guaranteed that in general, the Taylor series

converges everywhere.

• Suppose that the power series expansion converges. It

is not guaranteed that it converges to the function f.

Numerical examples

• Taylor series of sin(x) centred at x=1.

degree 1 degree 2 degree 3 degree 4