Symbolic Integration

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ismy tnh

Symbolic Integration

Presenter: Nguyn Qun

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Outline

1. Definition

2. Resultant

3. Integrals of Logarithmic and ExponentialExtension

4. Definite Integration

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Definition

1. Let R be an integral domain andD:RR such that

D(f+g) = D(f) + D(g) D(f*g) = D(f)*g + f*D(g)

D is called a differential operator

2. If f and g R and D(f) = g f is anintegral of g and f = g

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Definition

1. u = log(p) D(u) = D(p)/p

2. u = exp(p) D(u)/u = D(p)

3. u is called algebraic p(u) = 0 (p isa polynomial)

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Resultant

1. Let f (x) R[x] and g(x) R[x] beunivariate polynomials with real

coefficients. We want to determinewhether f and g have a common zero

2. We know already one technique for

solving the problem: Compute theGCD of f and g

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Resultant

1. We will see an alternative technique the resultant calculus.

2. In its basic form Only tell us whether f and g have a

common root

Will not tell us how many common roots Will not give a description of the

common roots

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Resultant

1. In this sense

Resultants are weaker than greatest

common divisors They are stronger to give us information

about common roots of multivariatepolynomials

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Resultant

1. Given polynomials f, g k[x] ofpositive degree, form:

f = a0xm + + am , a0 0 g = b0x

n + + bn , b0 0

2. We create the matrix (m+n)*(m+n)

as below, called Sylvester matrix orSyl(f,g,x)

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Resultant

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Resultant

1. The resultant of f and g, denoted byRes(f,g,x), is the determinant of the

Sylvester matrix Res(f,g,x)=det(Syl(f,g,x))

2. f and g have a common factor in k[x]

if and only if Res(f,g,x)=0

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Logarithmic and Exponential

Extension1. The algorithm

Hermite's method

Rothstein-Trager Method

2. Use these algorithm to calculate

Integrals of a Logarithmic Extension

Integrals of an Exponential Extension

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Hermite - Ostrogradsky's

algorithm1. Let f, g Z[x] be nonzero polynomials

deg(g) = n

deg(h) = m

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algorithm1. The result of the algorithm

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algorithm1. Hermite reduction find a, b, c, d, h

Q[x]

deg(a) < deg(b), deg(c) < deg(d) deg(b) + deg(d)


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Hermite - Ostrogradsky'salgorithm

1. In this formula h the polynomial part of the integral

c/d the rational part (a/b) the logarithm part

2. In practice

b = g* : the squarefree part of g d = g/g* a, c, h are uniquely determined

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Rothstein-Trager Theorem

1. Suppose our integral is

A(), B(): polynomials with coefficientsthat can be rational functions of x

deg A()


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Rothstein-Trager Theorem

1. The Rothstein-Trager Theorem says

R(z) be the resultant of B() and A()

z*(d B()/dx) with respect to 0 If all roots of R(z) are constants (r1, r2,

, rn) and

We have

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The Definite Integral

1. We use the expression to find thearea under a curve

2. F(x) is the integral of f(x)

3. F(b) is the value of the integral at theupper limit, x = b

4. F(a) is the value of the integral at thelower limit, x = a

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The Definite Integral

1. This expression is called a definiteintegral

2. It does not involve a constant ofintegration

3. It gives us a definite value (a number)

at the end of the calculation

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The Definite Integral Apps

1. In physics, workis done when a forceacting upon an object causes a

displacement. (For example, riding abicycle.)

2. IfF(x) is the variable force, to find the

work done, we use

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1. The average value of a function f(x)in the region x = a to x = b is given

by

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1. By using definite integral, we can findthe length of an arc along a curve

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References

http://www.apmaths.uwo.ca/~rcorless/AM563/NOTES/Nov_16_95/node8.html

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

http://math.rice.edu/~cbruun/vigre/vigreHW9.pdf http://www.intmath.com/integration/4-definite-

integral.php

http://www.mecca.org/~halfacre/MATH/appint.htm

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http://www.apmaths.uwo.ca/~rcorless/AM563/NOTES/Nov_16_95/node8.htmlhttp://www.apmaths.uwo.ca/~rcorless/AM563/NOTES/Nov_16_95/node8.htmlhttp://en.wikipedia.org/wiki/Symbolic_interactionismhttp://math.rice.edu/~cbruun/vigre/vigreHW9.pdfhttp://math.rice.edu/~cbruun/vigre/vigreHW9.pdfhttp://www.intmath.com/integration/4-definite-integral.phphttp://www.intmath.com/integration/4-definite-integral.phphttp://www.mecca.org/~halfacre/MATH/appint.htmhttp://www.mecca.org/~halfacre/MATH/appint.htmhttp://www.intmath.com/integration/4-definite-integral.phphttp://www.intmath.com/integration/4-definite-integral.phphttp://www.intmath.com/integration/4-definite-integral.phphttp://www.intmath.com/integration/4-definite-integral.phphttp://www.intmath.com/integration/4-definite-integral.phphttp://math.rice.edu/~cbruun/vigre/vigreHW9.pdfhttp://math.rice.edu/~cbruun/vigre/vigreHW9.pdfhttp://en.wikipedia.org/wiki/Symbolic_interactionismhttp://www.apmaths.uwo.ca/~rcorless/AM563/NOTES/Nov_16_95/node8.htmlhttp://www.apmaths.uwo.ca/~rcorless/AM563/NOTES/Nov_16_95/node8.html


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Questions and Answers

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Thank you

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Symbolic Integration