Strange behavior when replacing variables by numerical values [duplicate]

Question

I have a rather complicated function with parameters {a, b, c, d, e, f, k}, and I'd like to know its behavior as a function of k alone given other parameters, so I try the following code:

myFunction = D[ComplexExpand[Arg[((a^2 - 2 (2 k + I e - 2 c) (k + d - c))^2/((a^2 - 2 (2 k + I (b + e + 2 I c)) (k + d - c))^2 + 4 E^(2 I k f) b^2 (k + d - c)^2))], TargetFunctions -> {Re, Im}] // Simplify, k];
myFunctionWithValues[k_] := Evaluate[myFunction /. {b -> 10, a -> 2, c -> 100, d -> 0, e -> 1, f -> Pi/100} // FullSimplify]
LogPlot[myFunctionWithValues[k], {k, 95, 105}, PlotRange -> All]

and the outcome looks very bizarre. It can't be correct. To comfirm this, for example, I look at the value of myFunctionWithValues at the center k=100 by 1) directly replacing k by 100 and 2) by taking the limit k->100, and they disagree with each other:

myFunctionWithValues[k] /. k -> 100 // N

315120.

Limit[myFunctionWithValues[k], k -> 100] // N

10.

Apparently the latter gives a more reasonable result (which is checked by other approaches). Moreover, if I replace the value of all parameters (including k) in the very beginning, Mathematica also yields the same result:

myFunction /. {b -> 10, a -> 2, k -> c, d -> 0, e -> 1, f -> Pi/100} // Simplify

10

myFunction /. {b -> 10, a -> 2, c -> 100, d -> 0, e -> 1, f -> Pi/100, k -> 100} // N

10.

And the correct plot can be given by the following two ways:

(Limit[myFunctionWithValues[k], k -> #]) & /@ Range[95, 105, 1/10] //N;
ListPlot[%, PlotRange -> All, Joined -> True, AxesOrigin -> {100, 0}, DataRange -> {95, 105}, PlotMarkers -> Automatic]

Plot[myFunction /. {b -> 10, a -> 2, c -> 100, d -> 0, e -> 1, f -> Pi/100}, {k, 95, 105}, PlotRange -> All, AxesOrigin -> {100, 0}]

Question: What's going on here? Why can't I simply plot myFunctionWithValues but have to take the limit of k at each point? More precisely, what's the difference between using the replacement rules and taking limits? I thought it should be the same given that in my case there's no singularity causing trouble, otherwise Mathematica would give me an error message.

Answer 1

I'd define your functions with all the arguments :

myFunction[a_, b_, c_, d_, e_, f_, k_] = D[ComplexExpand[
  Arg[((a^2 - 2 (2 k + I e - 2 c) (k + d - c))^2/((a^2 - 2 (2 k + I (b + e + 2 I c)) 
  (k + d - c))^2 + 4 E^(2 I k f) b^2 (k + d - c)^2))], 
 TargetFunctions -> {Re, Im}] // Simplify, k];

myFunctionWithValues[k_] = myFunction[2, 10, 100, 0, 1, Pi/100, k];

Plot[{myFunctionWithValues[k], Log[myFunctionWithValues[k]]}, {k, 95, 105},         
   PlotRange -> All]