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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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Hermite - Ostrogradsky's
algorithm1. The result of the algorithm
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Hermite - Ostrogradsky's
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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The Definite Integral Apps
1. The average value of a function f(x)in the region x = a to x = b is given
by
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The Definite Integral Apps
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.html8/2/2019 Symbolic Integration
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Questions and Answers
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Thank you
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