Math 2.0

Integral transforms, operational calculus Mikusinski calculus and other operational calculi Proposition

Differentiation operator as inverse of integration

Builds on Mikusiński field of operatorsIntegration operator as convolution with the unit stepFundamental theorem of calculus

Statement

Let ℳ\mathcal{M} be the Mikusiński field of operators, the field of fractions of the convolution ring of locally integrable functions on [0,∞)[0, \infty). Let hh be the constant function 11 on [0,∞)[0, \infty), and let s=1/hs = 1/h be its inverse in ℳ\mathcal{M}. For a complex number cc, write cc also for the numerical operator c⋅1c \cdot 1 of ℳ\mathcal{M}, which corresponds to the function c hc\,h divided by hh. Let ff be continuously differentiable on [0,∞)[0, \infty). Then sf=f′+f(0)s f = f' + f(0) in ℳ\mathcal{M}. More generally, if ff is nn times continuously differentiable on [0,∞)[0, \infty), then snf=f(n)+∑k=0n−1f(k)(0) sn−1−k.s^n f = f^{(n)} + \sum_{k=0}^{n-1} f^{(k)}(0) \, s^{n-1-k}.

Remarks

By the fundamental theorem of calculus, ∫0tf′(τ) dτ=f(t)−f(0)\int_0^t f'(\tau) \, d\tau = f(t) - f(0). By the integration operator result this reads hf′=f−f(0)hh f' = f - f(0) h in the convolution ring. Multiplying both sides by ss gives f′=sf−f(0)f' = s f - f(0), which is the first formula. The general formula follows by induction on nn. For example, with f(t)=eatf(t) = e^{a t}, one gets sf=af+1s f = a f + 1, so f=1/(s−a)f = 1/(s - a) in ℳ\mathcal{M}. The operator ss is not a function: it is a quotient of functions, and sfs f equals the function f′f' only when f(0)=0f(0) = 0.

Going deeper

For a student

This formula has the same shape as the rule ℒ{f′}(s)=sF(s)−f(0)\mathcal{L}\{f'\}(s) = s F(s) - f(0) for the Laplace transform. Mikusiński's calculus gets it with no condition on the growth of ff and no convergence of integrals. A linear differential equation with constant coefficients and initial values becomes an algebraic equation in ss in the field ℳ\mathcal{M}. One then solves for the unknown and reads off a function, often after a partial fraction expansion in ss.

Reading path

To understand this page from the ground up, read these 43 pages in order. The foundations come first.

  1. Convolution on the half-line concept
  2. Function concept
  3. Limit point of a set definition
  4. Local maximum definition
  5. Mikusiński field of operators concept
  6. Partition of an interval definition
  7. Sequence definition
  8. Laplace transform definition
  9. Limit of a function definition
  10. Local minimum definition
  11. Mesh of a partition definition
  12. Riemann sum definition
  13. Real sequence definition
  14. Subsequence definition
  15. Heaviside step function object
  16. Continuity at a point definition
  17. Derivative definition
  18. Riemann integrable function definition
  19. Bounded sequence definition
  20. Limit of a sequence definition
  21. Monotone sequence definition
  22. Integration operator as convolution with the unit step statement
  23. Continuous function definition
  24. Antiderivative definition
  25. Fermat's theorem on stationary points statement
  26. Riemann integral definition
  27. Convergent sequence definition
  28. Monotone convergence theorem for sequences statement
  29. Every sequence has a monotone subsequence statement
  30. Intermediate value theorem statement
  31. Definite integral definition
  32. Bolzano–Weierstrass theorem statement
  33. Additivity of the integral over intervals statement
  34. Boundedness theorem statement
  35. Monotonicity of the integral statement
  36. Extreme value theorem statement
  37. Mean value theorem for integrals statement
  38. Rolle's theorem statement
  39. First fundamental theorem of calculus statement
  40. Mean value theorem statement
  41. Second fundamental theorem of calculus statement
  42. Fundamental theorem of calculus statement
  43. Differentiation operator as inverse of integration statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Operational_calculus
About this page
TagDBM13YSJAM
URIm2://DBM13YSJAM
Kindstatement, proposition
WritingW0 stub
MSC44A40
StatusThis copy of the site has no build data. Run m2 status DBM13YSJAM.
AboutMikusiński field of operators, Mikusiński operational calculus
UsesMikusiński field of operators, Integration operator as convolution with the unit step, Fundamental theorem of calculus

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