Consider the following integral:
FullSimplify[Integrate[r BesselJ[m,BesselJZero[m,n]r /a]^2,{r,0,a}],
Assumptions->n\[Element]Integers&&n>=1&&m\[Element]Integers]
(* ConditionalExpression[1/2 a^2 BesselJ[-1+m,BesselJZero[m,n]]^2,m>-1] *)
This gives part of the orthogonality condition of BesselJ
functions (see here, Eq. 53).
What I can't figure out is the reason for the ConditionalExpression
. Mathematica knows about the relationship:
FullSimplify[BesselJ[-m,x]==(-1)^m BesselJ[m,x],Assumptions->m\[Element]Integers]
(* True *)
and can even calculate the integral given m=-1
, for example, explicitly:
Integrate[r BesselJ[-1, BesselJZero[-1, n] r /a]^2, {r, 0, a}]
(* 1/2 a^2 (BesselJ[0,BesselJZero[-1,n]]^2+BesselJ[1,BesselJZero[-1,n]]^2-
(2 BesselJ[0,BesselJZero[-1,n]] BesselJ[1,BesselJZero[-1,n]])/BesselJZero[-1,n]) *)
But, trying to force Integrate
to assume m<=-1
throws an error:
Integrate[r BesselJ[m,BesselJZero[m,n]r /a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&n>=1&&m\[Element]Integers&&m<=-1]
(* Integrate::idiv: Integral of r BesselJ[m,(r BesselJZero[m,n])/a]^2
does not converge on {0,a}.*)
So, why can't it solve it for any (integer) m
in the first (or negative integers in the last) scenario?
Edit Additional strangeness based on belisarius's comment. If you directly integrate something like:
Integrate[r BesselJ[15,BesselJZero[15,n] r/a]^2,{r,0,a},Assumptions->n\[Element]Integers&&n>=1]
it outputs a giant monstrosity. Yet, this monstrosity is (nearly) equivalent to the smaller expression:
N[(% /. n -> 1)]-
N[(1/2 a^2 BesselJ[-1 + 15, BesselJZero[15, n]]^2 /. n -> 1)]
(* 6.12357*10^-16 a^2 *)
It seems that it uses a different method for integrating these functions with an explicit value for m
than without. And for some reason doesn't use the (-1)^m
BesselJ
identity for solving the m<=-1
cases.
Edit 2 Based on J.M.'s answer this gets stranger. First of all, integrating:
Integrate[r BesselJ[m,BesselJZero[m,n] r/a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&Positive[n]&&m\[Element]Integers&&Negative[m]]
works fine. As does,
Integrate[r BesselJ[m,BesselJZero[m,n] r/a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&n>=1&&m\[Element]Integers&&Negative[m]]
and even
Integrate[r BesselJ[m,BesselJZero[m,n] r/a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&n>=1&&m\[Element]Integers&&m<0]
But,
Integrate[r BesselJ[m,BesselJZero[m,n] r/a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&n>=1&&m\[Element]Integers&&m<=-1]
fails with the error from above.
Furthermore, even though
Integrate[r BesselJ[m,BesselJZero[m,n] r/a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&n>=1&&m\[Element]Integers&&Negative[m]]==
Integrate[r BesselJ[m,BesselJZero[m,n] r/a]^2,{r,0,a},
Assumptions->n\[Element]Integers&&Positive[n]&&m\[Element]Integers&&Positive[m]]
(* True *)
the timing to solve these are way different. With
Integrate[ snip Positive[m]];//Timing
Integrate[ snip Negative[m]];//Timing
(* {15.538, Null}
{0.421, Null} *)
which is a huge increase in time for essentially the same integral. Any further ideas?
Integrate[r BesselJ[m, BesselJZero[m, n] r/a]^2, {r, 0, a}, Assumptions -> n \[Element] Integers && n >= 1 && m \[Element] Integers && m == -15]
and take a look at those coefficients $\endgroup$m==15
. Yet, it can also be expressed as1/2 a^2 BesselJ[-1 + 15, BesselJZero[15, n]]^2
. $\endgroup$