Math 2.0

Approximations and expansions Definition

Padé table

Builds on Padé approximantFormal power seriesTaylor polynomial

Definition

Let f(z)=∑k≥0ckzkf(z) = \sum_{k \ge 0} c_k z^k be a formal power series with complex coefficients. For integers m,n≥0m, n \ge 0 let [m/n]f[m/n]_f be the Padé approximant of ff with numerator degree at most mm and denominator degree at most nn. The Padé table of ff is the doubly infinite array ([m/n]f)m,n≥0\big([m/n]_f\big)_{m, n \ge 0}. The table is normal if no rational function occurs at two different positions (m,n)(m, n).

Remarks

The column n=0n = 0 consists of the Taylor polynomials of ff. By a theorem of Padé, the positions where a given rational function occurs form a square block, possibly infinite. Example: for f=ezf = e^z the table is normal, and [1/1]f=(2+z)/(2−z)[1/1]_f = (2 + z)/(2 - z). Non-example of normality: for f=1/(1−z)=∑zkf = 1/(1 - z) = \sum z^k, one has [m/n]f=1/(1−z)[m/n]_f = 1/(1 - z) for all m≥0m \ge 0 and n≥1n \ge 1, since (1−z)f−1=0(1 - z) f - 1 = 0; so this table is not normal. Orientation conventions differ: some authors put mm along rows and nn along columns, others the reverse.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

Show the path as a graph
Formal power series, definition, no evidence levelFormal power seriesdefinitionGroup, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, 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 pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementDifferentiable function, definition, no evidence levelDifferentiable functi…definitionConvergent 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 continuitydefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementHigher derivative, definition, no evidence levelHigher derivativedefinitionLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionProduct 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 functiondefinitionPadé approximant, definition, no evidence levelPadé approximantdefinitionPadé table, definition, no evidence levelPadé tabledefinition
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. Formal power series definition
  2. Group concept
  3. Limit point of a set definition
  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. Derivative definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Degree of a polynomial definition
  14. Continuous function definition
  15. Epsilon-delta definition of continuity statement
  16. Differentiable function definition
  17. Convergent sequence definition
  18. Epsilon-N definition of the limit of a sequence statement
  19. Sequential continuity definition
  20. Derivative function definition
  21. Continuity is equivalent to sequential continuity statement
  22. Convergent sequences are bounded statement
  23. Higher derivative definition
  24. Limit of a product of sequences statement
  25. Taylor polynomial definition
  26. Product of continuous functions is continuous statement
  27. Limit of a quotient of sequences statement
  28. Polynomial functions are continuous statement
  29. Quotient of continuous functions is continuous statement
  30. Rational function definition
  31. Padé approximant definition
  32. Padé table definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Pad%C3%A9_table
About this page
TagKE1KD9P1J0
URIm2://KE1KD9P1J0
Kinddefinition
WritingW0 stub
MSC41A21, 30B70, 30E10
DefinesPadé table
UsesPadé approximant, Formal power series, Taylor polynomial

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.