Integral of a function defined with a loop

Question

I have defined a function g as
g[t_] := (
   res = 0;
   i = 1;
   While[i <= t,
     res = res + i;
     i = i + 1;
   ];
   res);

The aim is to work with the function F[u], which should be the integral of g in bounds $[0,u]$, something like
F[u_] := Integrate[g[y], {y, 0, u}]

However, the result I obtain for F is not correct with such definition of g. In fact, F takes value 0 for any argument u (my guess is that this happens because g[y] is immediately evaluated as 0).
How can F be redefined properly, without changing the definition of g?

Bob Hanlon · Answer

Clear["Global`*"] Use Module to keep the temporary variable names (and values) out of the Global context. g[t_] := Module[{res = 0, i = 1}, While[i <= t, res = res + i; i = i + 1]; res]; However, note that g evaluates to 0 for symbolic arguments. g[t] (* 0 *) Consequently, restrict its arguments to being NumericQ or preferably, Positive. Clear[g] g[t_?Positive] := Module[{res = 0, i = 1}, While[i <= t, res = res + i; i = i + 1]; res]; For comparison purposes, define g2[t_] := Module[{m = Floor[t]}, m (m + 1)/2] The argument of g2 does not need to be restricted. g2[t] (* 1/2 Floor[t] (1 + Floor[t]) *) Numerically, g and g2 are equivalent for positive arguments. Plot[{g[t], g2[t]}, {t, 0, 10}, PlotStyle -> {Automatic, Dashed}, PlotLegends -> Placed["Expressions", {.25, .75}], Exclusions -> Range[10]] The integrals are F[u_?NumericQ] := Integrate[g[y], {y, 0, u}] F2[u_?NumericQ] := Integrate[g2[y], {y, 0, u}] Plot[{F[u], F2[u]}, {u, 0, 5}, PlotStyle -> {Automatic, Dashed}, PlotLegends -> Placed["Expressions", {.25, .75}]] // Quiet

Integral of a function defined with a loop

One Answer

Add your own answers!

Ask a Question