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I have a function of the following form: $$ \varphi_n(x)= \underbrace{\int\limits_\mathbb{R}\ldots\int\limits_\mathbb{R}}_{n}\exp\left[-\sum\limits_{j=2}^{n}\left(x_{j}-x_{j-1}\right)^2-(x-x_{n})^2\right]dx_1\ldots dx_{n}\quad \left(\varphi_n(x)\equiv \pi^{n/2}\right), $$ and I'm trying to define a function Phi[x, n] that will calculate the above integral. I know, how to define this function for fixed $n$, for example $$ \varphi_1(x)= \int\limits_\mathbb{R}e^{-(x-x_{1})^2}dx_1, $$ defines as

Phi1 = Integrate[Exp[-(x - #)^2], {x, -Infinity, Infinity}] &;

and $$ \varphi_2(x)= \int\limits_\mathbb{R}\int\limits_\mathbb{R}e^{-(x_2-x_1)^2-(x-x_2)^2}dx_1dx_2, $$ defines as

Phi2 = Integrate[Integrate[Exp[-(x2 - x1)^2 - (# - x2)^2], {x1, -Infinity, Infinity}], {x2, -Infinity, Infinity}] &;

How to define a function, that would take $n$ as an argument?

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I bet you could do this more elegantly with Fold, but this gets you there,

Phi[n_] := 
 Integrate[Exp[-Sum[(x[j] - x[j - 1])^2, {j, 2, n}] - (x - x[n])^2], 
  Sequence @@ ({x[#], -∞, ∞} & /@ Range[n])]

Phi /@ Range[5]
(* {Sqrt[π], π, π^(3/2), π^2, π^(5/2)} *)

You could make it a function of x, but as you write, the answer is independent of x so I don't see why you would.

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  • $\begingroup$ @Glinka In order to obtain a true dependence on x you could change the integration interval. (Although the integral might not be solved symbolically anymore). $\endgroup$ – Dr. Wolfgang Hintze Mar 16 '16 at 15:47
  • $\begingroup$ @JasonB, thank you for your answer! Your method worked on the example from the question, and it worked when I used a function that actually depended on $x$. The problem though that it works very slow when $n>3$ (for $n=3$ the process lasts several minutes and for $n=4$ the process doesn't seem to end). I figured that I use NIntegrate instead of Integrate, it would speed up a process. But when I just change Integrate to NIntegrate in your code, it show the following error: Invalid integration variable or limit(s) in {x[1],-[Infinity],[Infinity]}. What do I do wrong? $\endgroup$ – Glinka Mar 16 '16 at 19:03
  • $\begingroup$ @Glinka It's true that multidimensional integrals can take a while, and can be sped up by using numerical methods. But it's hard to fix an error without having the code to reproduce it. You could post it here as a comment or as another question. $\endgroup$ – Jason B. Mar 16 '16 at 19:07
  • $\begingroup$ @Dr.WolfgangHintz, in our task we always consider integrals over $\mathbb{R}$, and a true dependence on x occurs when we consider more complex integrands. In the example from the question $x$ variable is dummy by accident. $\endgroup$ – Glinka Mar 16 '16 at 19:12
  • $\begingroup$ @JasonB, I posted it as a new question: mathematica.stackexchange.com/questions/110197/… $\endgroup$ – Glinka Mar 16 '16 at 19:36
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My take:

With[{n = 6}, 
     Integrate[Exp[-#.# & @ Differences[Append[Array[C, n], x]]], ##] & @@ 
     Array[{C[#], -∞, ∞} &, n]]
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Here's some silliness along the lines of Jason B's answer:

int[n_]:= Integrate[
  Exp[-#^2] &@*Subtract @@@ Partition[Array[x, n + 1, 0], 2, 1] // Times @@ # &, 
  Sequence @@ Array[{x[#], -∞, ∞} &, 3]
 ]
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