Explore Continuous and Discrete Calculus Analogs
Explore some of the many analogs that exist between continuous and discrete calculus operations.
 In[1]:= ```grid = Grid[{ {"Derivative", "\!\(\*SubscriptBox[\"\[PartialD]\", \"x\"]\)u[x]", "Difference", "\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \"x\"]\)u[x]"}, {"Indefinite Integral", "\[Integral]u[x]\[DifferentialD]x", "Indefinite Sum", "\!\(\*SubscriptBox[\"\[Sum]\", \"x\"]\)u[x]"}, {"Definite Integral", "\!\(\*SubsuperscriptBox[\"\[Integral]\", \"a\", \"b\"]\)u[x]\ \[DifferentialD]x", "Definite Sum", "\!\(\*UnderoverscriptBox[\"\[Sum]\", RowBox[{\"k\", \"=\", \"a\"}], \"b\"]\)u[x]"}, {"Differential \ Equation", "F[u[x],\!\(\*SubscriptBox[\"\[PartialD]\", \"x\"]\)u[x],\ \[Ellipsis],\!\(\*SubscriptBox[\"\[PartialD]\", RowBox[{\"{\", RowBox[{\"x\", \",\", \"n\"}], \"}\"}]]\)u[x]]\[Equal]0]", "Difference Equation", "F[u[x],\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \"x\"]\)u[x],\ \[Ellipsis],\!\(\*SubscriptBox[\"\[DifferenceDelta]\", RowBox[{\"{\", RowBox[{\"x\", \",\", \"n\"}], \"}\"}]]\)u[x]]\[Equal]0]"}, {"Exponential", "\!\(\*SubscriptBox[\"\[PartialD]\", \ \"x\"]\)\!\(\*SuperscriptBox[\"\[ExponentialE]\", \ \"x\"]\)=\!\(\*SuperscriptBox[\"\[ExponentialE]\", \"x\"]\)", "Exponential", "\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \ \"x\"]\)\!\(\*SuperscriptBox[\"2\", \"x\"]\)=\!\(\*SuperscriptBox[\"2\ \", \"x\"]\)"}, {"Logarithm", "\!\(\*SubscriptBox[\"\[PartialD]\", \ \"x\"]\)Log[x]=\!\(\*FractionBox[\"1\", \"x\"]\)", "PolyGamma", "\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \ \"x\"]\)PolyGamma[x]=\!\(\*FractionBox[\"1\", \"x\"]\)"}, \ {"Linearity", "\!\(\*SubscriptBox[\"\[PartialD]\", \ \"x\"]\)(u[x]+v[x])=\!\(\*SubscriptBox[\"\[PartialD]\", \"x\"]\)u[x]+\ \!\(\*SubscriptBox[\"\[PartialD]\", \"x\"]\)v[x]", "Linearity", "\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \"x\"]\)(u[x]+v[x])=\ \!\(\*SubscriptBox[\"\[DifferenceDelta]\", \ \"x\"]\)u[x]+\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \ \"x\"]\)v[x]"}, {"Power", "\!\(\*SuperscriptBox[\"x\", \"n\"]\)", "FactorialPower", "\!\(\*TemplateBox[{\"x\",\"n\"},\n\"FactorialPower\"]\)"}, {"Integral of a Power", "\[Integral]\!\(\*SuperscriptBox[\"x\", \ \"n\"]\)\[DifferentialD]x=\!\(\*FractionBox[SuperscriptBox[\"x\", RowBox[{\"n\", \"+\", \"1\"}]], RowBox[{\"n\", \"+\", \"1\"}]]\)", "Sum of a Factorial Power", "\!\(\*SubscriptBox[\"\[Sum]\", \ \"x\"]\)\!\(\*TemplateBox[{\"x\",\"n\"},\n\ \"FactorialPower\"]\)=\!\(\*FractionBox[TemplateBox[{\"x\",RowBox[{\"\ n\", \"+\", \"1\"}]},\n\"FactorialPower\"], RowBox[{\"n\", \"+\", \"1\"}]]\)"}, {"Power Series", "f[x] = \!\(\*SubsuperscriptBox[\"\[Sum]\", RowBox[{\"k\", \"=\", \"0\"}], \"\[Infinity]\"]\)\!\(\*FractionBox[ RowBox[{SuperscriptBox[\"\[PartialD]\", \"k\"], RowBox[{\"f\", \"[\", \"a\", \"]\"}]}], RowBox[{\"k\", \"!\"}]]\)(x-a\!\(\*SuperscriptBox[\")\", \"k\"]\)", "Factorial Power Series", "f[x] = \!\(\*SubsuperscriptBox[\"\[Sum]\", RowBox[{\"k\", \"=\", \"0\"}], \"\[Infinity]\"]\)\!\(\*FractionBox[ RowBox[{SuperscriptBox[\"\[DifferenceDelta]\", \"k\"], RowBox[{\"f\", \"[\", \"a\", \"]\"}]}], RowBox[{\"k\", \"!\"}]]\)(x-a\!\(\*SuperscriptBox[\")\", RowBox[{\"(\", \"k\", \")\"}]]\)"}, {"Fundamental Theorem", "\[Integral]\!\(\*SubscriptBox[\"\[PartialD]\", \"x\"]\)f[x]\ \[DifferentialD]x=f[x]+C", "Fundamental Theorem", "\!\(\*SubscriptBox[\"\[Sum]\", \"x\"]\)\!\(\*SubscriptBox[\"\ \[DifferenceDelta]\", \"x\"]\)f[x]=f[x]+C"}, {"Integration by Parts", "\[Integral]f[x]\!\(\*SubscriptBox[\"\[PartialD]\", \ \"x\"]\)g[x]\[DifferentialD]x \[Equal] \ f[x]g[x]-\[Integral]\!\(\*SubscriptBox[\"\[PartialD]\", \ \"x\"]\)f[x]g[x]\[DifferentialD]x", "Summation by Parts", "\!\(\*SubscriptBox[\"\[Sum]\", \ \"x\"]\)f[x]\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \ \"x\"]\)g[x]=f[x]g[x]-\!\(\*SubscriptBox[\"\[Sum]\", \ \"x\"]\)\!\(\*SubscriptBox[\"\[DifferenceDelta]\", \ \"x\"]\)f[x]\!\(\*TemplateBox[{RowBox[{\"g\", \"[\", \"x\", \ \"]\"}],\"x\"},\n\"DiscreteShift2\"]\)"}, {"Laplace Transform", "\!\(\*SubsuperscriptBox[\"\[Integral]\", \"0\", \ \"\[Infinity]\"]\)f[x]\!\(\*SuperscriptBox[\"\[ExponentialE]\", RowBox[{ RowBox[{\"-\", \" \", \"s\"}], \" \", \"x\"}]]\)\[DifferentialD]x", "Z Transform", "\!\(\*UnderoverscriptBox[\"\[Sum]\", RowBox[{\"n\", \"=\", \"0\"}], \ \"\[Infinity]\"]\)f[n]\!\(\*SuperscriptBox[\"z\", RowBox[{\"-\", \"n\"}]]\)"}, {"Fourier Transform", "\!\(\*FractionBox[\"1\", SqrtBox[ RowBox[{\"2\", \" \", \"\[Pi]\"}]]]\)\!\(\*SubsuperscriptBox[\"\ \[Integral]\", RowBox[{\"-\", \"\[Infinity]\"}], \ \"\[Infinity]\"]\)f[x]\!\(\*SuperscriptBox[\"\[ExponentialE]\", RowBox[{\"\[ImaginaryI]\", \" \", \"t\", \" \", \"x\"}]]\)\ \[DifferentialD]x", "Discrete-Time Fourier Transform", "\!\(\*UnderoverscriptBox[\"\[Sum]\", RowBox[{\"n\", \"=\", RowBox[{\"-\", \"\[Infinity]\"}]}], \ \"\[Infinity]\"]\)f[n]\!\(\*SuperscriptBox[\"\[ExponentialE]\", RowBox[{ RowBox[{\"-\", \"\[ImaginaryI]\"}], \" \", \"n\", \" \", \ \"\[Omega]\"}]]\)"}}, Dividers -> All, Alignment -> {{Left, Center, Left, Center}, Baseline}, Background -> LightYellow, Spacings -> {1, 2}, ItemSize -> Full];```
 In[2]:= ```TraditionalForm@ Style[grid, {FontFamily -> "Helvetica", FontSize -> Small}]```