Consider the Mathieu characteristic function, which is a piecewise function. The discontinuity happens at integer number.
With[{V0 = -1}, Plot[MathieuCharacteristicA[κ, V0], {κ, -2.5, 2.5}]]
I'm trying to calculate the limit at k=2 with the direction of positive axis. I tried to do this in Mathematic and I got different answers when approaching 2:
So I calculate the function value at 2-ϵ, and with ϵ take to be smaller and smaller values. We can see that the function value first become small, then at ϵ=10^-60, the function value suddenly jumps about 40 orders of magnitude.
N[
Table[Re@MathieuC[MathieuCharacteristicA[2 - ϵ, -1], -1,
0], {ϵ, {10^-6, 10^-8, 10^-10, 10^-20, 10^-40, 10^-60,
10^-100}}], 100]
(*
9.37551974147072835559249118350860328663842780156187022041630631583395\
1776806837902179623867179198570*10^-6, \
9.37551974249387199028571957345692499510682012392156540333196700968792\
3231481580841077469030562191305*10^-8, \
9.37551974249397430464920759145123255395714887241690706549073245295310\
8780592926489536671069329119067*10^-10, \
9.37551974249397431488166718533639787521652887810091370096758925579343\
2748631135267810942389900367393*10^-20, \
9.37551965486425358591047481958041604234458729394544036776955645793908\
5038181962308922619162634749446*10^-40, \
1.11438859178173311502100242852087617176800818468416114397816239922310\
7768160400228809114697590700165, \
1.11438859178173311502100242852087617176800818468416114397816239922310\
7768160400228809114697590700165
*)
So I'm wondering whether this is the behavior of Mathieu function, or simply some kind of error in Mathematica. If we look at the behavior before ϵ decrease to 10^-60, it looks like the function limit is 0. So is 0 the correct answer for the limit of MathieuC[MathieuCharacteristicA[2 - ϵ, -1]?