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DREAM. IDEA. PLAN. IMPLEMENTATION. Introduction to Image Processing. Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University. Dr. Kourosh Kiani Email: kkiani2004@yahoo.com Email: Kourosh.kiani@aut.ac.ir Email : Kourosh.kiani@semnan.ac.ir
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DREAM IDEA PLAN IMPLEMENTATION
Introduction to Image Processing Present to:Amirkabir University of Technology (Tehran Polytechnic) & Semnan University Dr. Kourosh Kiani Email: kkiani2004@yahoo.com Email: Kourosh.kiani@aut.ac.ir Email: Kourosh.kiani@semnan.ac.ir Web: www.kouroshkiani.com
Lecture 06 TAYLOR SERIES
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. Brook Taylor 1685 - 1731 Greg Kelly, Hanford High School, Richland, Washington
How it works ? f(x) = exp(x) Power Series
How it works ? Power Series f(x) = ex f(x) = x0
How it works ? Power Series f(x) = ex f(x) = 1 + x
How it works ? Power Series f(x) = ex f(x) = 1 + x +
How it works ? Power Series f(x) = ex f(x) = 1 + x +
Power Series When it is used ? Estimation of values of transcendental functions Plotting of an ”ugly” function Estimation of integrals Aproximate solving of differential equations Modelling
Power Series Problems Convergence problems ex 1+x+x2/2+x3/6 Singular functions
If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation. Suppose we wanted to find a fourth degree polynomial of the form: that approximates the behavior of at
If we plot both functions, we see that near zero the functions match very well!
This pattern occurs no matter what the original function was! Our polynomial: has the form: or:
Maclaurin Series: (generated by f at ) Taylor Series: (generated by f at ) If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:
example: Rather than start from scratch, we can use the function that we already know:
There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet.
The 3rd order polynomial for is , but it is degree 2. When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 terms. It is a 5th order Taylor polynomial, because it was found using the 4th derivative. It is also a 4th degree polynomial, because x is raised to the 4th power.
List the function and its derivatives. Evaluate column one for x = 0. This is a geometric series with a = 1 and r = x.
We could generate this same series for with polynomial long division:
This is a geometric series with a = 1 and r = -x.
We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly. They do help to explain where the formula for the sum of an infinite geometric comes from. We will find other uses for these series, as well. A more impressive use of Taylor series is to evaluate transcendental functions.
Both sides are odd functions. Sin (0) = 0 for both sides.
and substitute for , we get: If we start with this function: This is a geometric series with a = 1 and r = -x2. If we integrate both sides: This looks the same as the series for sin (x), but without the factorials.
An amazing use for infinite series: Substitute xi for x. Factor out the i terms.
This is the series for sine. This is the series for cosine.
Taylor Series - example Find the Taylor series for f(x) = sin(x) based on an expansion around a=/6
