Skip to main content

calculus and analysis - Integrating hat functions



I am trying to integrate a hat function for a project that I am doing and have found a method to do so but I find it sloppy. Currently I have the basis function


\[Psi][z_] := z - Subscript[Z, i]/ \[CapitalDelta]z + 1;

which I am trying to integrate from $z_{i-1}$ to $z_{i+1}$. I break the basis function up into two pieces and integrate the left side from $z_{i-1}$ to $z_{i}$ and then the right side from $z_i$ to $z_{i+1}$. My first question is, is there a way to integrate piecewise functions? The second question I have is, is there a way to set global assumptions like $z_{i-1} < z_i < z_{i+1}$, $z_i - z_{i-1} = \Delta z$ , etc?


Edit: This is the piece wise function taken directly from my code I am trying to integrate


\[Psi][z_, c_] :=  Piecewise[{{(z - c)/\[CapitalDelta]z + 1, 
 z <= c}, {-(z - c)/\[CapitalDelta]z + 1, z > c}}];

where $c$ is the center of the hat function. Here is my attempt to integrate the piece wise function


 FullSimplify[

 Integrate[\[Psi][z, Subscript[Z, i]], {z, Subscript[Z, i - 1], 
   Subscript[Z, i + 1]}], 
 Assumptions -> {-(Subscript[Z, i + 1] - Subscript[Z, 
       i ]) == -\[CapitalDelta]z, 
   Subscript[Z, i + 1] - Subscript[Z, 
     i ] == \[CapitalDelta]z, -(Subscript[Z, i] - Subscript[Z, 
       i - 1 ]) == -\[CapitalDelta]z, 
   Subscript[Z, i] - Subscript[Z, i - 1 ] == \[CapitalDelta]z}]

I do not get a usable answer. Am I doing something wrong (ie can one integrate a piece wise function)?




Answer



Short answer: Mathematica has no problem integrating piecewise or hat functions.


Your notation seems to me to be needlessly complex. Why bother to define $Z_i$ when it's just $Z_0+i\Delta z$? Isn't your $\psi$ just 1-Abs[z-c]/Δz? However, I'll try to adhere to the spirit of your notation. Here's some code that works for me:


Zdefs = {Subscript[Z, i] - Subscript[Z, i - 1] == Δz, 
   Subscript[Z, i + 1] - Subscript[Z, i] == Δz};
Integrate[ψ[z, Subscript[Z, i]],
   {z, Subscript[Z, i - 1], Subscript[Z, i + 1]} /. 
  Solve[Zdefs, {Subscript[Z, i + 1], Subscript[Z, i - 1]}][[1]], 
 Assumptions -> Δz > 0 && 
   Subscript[Z, i] ∈ Reals]

Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more