Math 2.0

Real functions Definition

Heaviside cover-up method

Builds on Rational functionPartial fraction decompositionRoot of a polynomial

Definition

Let PP and QQ be complex polynomials with deg⁡P<deg⁡Q\deg P < \deg Q, and let F=P/QF = P/Q be the rational function. Let aa be a root of QQ of multiplicity m≥1m \ge 1, and write Q(s)=(s−a)mR(s)Q(s) = (s - a)^m R(s) with RR a polynomial and R(a)≠0R(a) \ne 0. In the partial fraction decomposition of FF, let cc be the coefficient of 1/(s−a)m1/(s - a)^m. The Heaviside cover-up method computes this coefficient as c=P(a)R(a)=[(s−a)mF(s)]s=a.c = \frac{P(a)}{R(a)} = \Big[ (s - a)^m F(s) \Big]_{s = a}. When aa is a simple root, this equals P(a)/Q′(a)P(a)/Q'(a).

Remarks

The formula holds because (s−a)mF(s)(s - a)^m F(s) equals cc plus terms that vanish at s=as = a. Example: for F(s)=1(s−1)(s−2)(s−3)F(s) = \frac{1}{(s - 1)(s - 2)(s - 3)}, cover-up gives the coefficients 1(1−2)(1−3)=12\frac{1}{(1-2)(1-3)} = \tfrac12, 1(2−1)(2−3)=−1\frac{1}{(2-1)(2-3)} = -1 and 1(3−1)(3−2)=12\frac{1}{(3-1)(3-2)} = \tfrac12. Its inverse Laplace transform is therefore 12et−e2t+12e3t\tfrac12 e^{t} - e^{2t} + \tfrac12 e^{3t}. Non-example: for a repeated root the method gives only the top coefficient. For 1s(s−1)2=As+Bs−1+C(s−1)2\frac{1}{s (s - 1)^2} = \frac{A}{s} + \frac{B}{s - 1} + \frac{C}{(s - 1)^2}, cover-up gives A=1A = 1 and C=1C = 1. The true value B=−1B = -1 does not come from covering up; one finds it by comparing coefficients or by differentiating (s−1)2F(s)(s - 1)^2 F(s) at s=1s = 1.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionPartial fraction decomposition, concept, no evidence levelPartial fraction deco…conceptSequence, definition, no evidence levelSequencedefinitionRing, definition, no evidence levelRingdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…statementRational function, definition, no evidence levelRational functiondefinitionHeaviside cover-up method, definition, no evidence levelHeaviside cover-up me…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Limit point of a set definition
  3. Partial fraction decomposition concept
  4. Sequence definition
  5. Ring definition
  6. Limit of a function definition
  7. Real sequence definition
  8. Polynomial ring definition
  9. Continuity at a point definition
  10. Bounded sequence definition
  11. Limit of a sequence definition
  12. Polynomial evaluation definition
  13. Continuous function definition
  14. Epsilon-delta definition of continuity statement
  15. Convergent sequence definition
  16. Epsilon-N definition of the limit of a sequence statement
  17. Sequential continuity definition
  18. Root of a polynomial definition
  19. Continuity is equivalent to sequential continuity statement
  20. Convergent sequences are bounded statement
  21. Limit of a product of sequences statement
  22. Product of continuous functions is continuous statement
  23. Limit of a quotient of sequences statement
  24. Polynomial functions are continuous statement
  25. Quotient of continuous functions is continuous statement
  26. Rational function definition
  27. Heaviside cover-up method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Heaviside_cover-up_method
About this page
TagM2FSQNBJCT
URIm2://M2FSQNBJCT
Kinddefinition
WritingW0 stub
MSC26C15, 44A10
DefinesHeaviside cover-up method
UsesRational function, Partial fraction decomposition, Root of a polynomial

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