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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?

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try 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 –  belisarius Jul 6 '12 at 18:49
    
@belisarius But you get the same mess using m==15. Yet, it can also be expressed as 1/2 a^2 BesselJ[-1 + 15, BesselJZero[15, n]]^2. –  Eli Lansey Jul 6 '12 at 19:01
    
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2 Answers 2

up vote 3 down vote accepted

While trying to investigate this, I hit upon the following integral:

Integrate[BesselJ[Floor[m], r]^2, {r, 0, 1}, Assumptions -> m <= -1]

0

This is obviously an incorrect result. It only appears when I add the assumption above. This leads me to suspect that the failure of Eli's integral in the same range of m is related to this - and I'd call it a bug that needs to be reported.

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Apparently, what you should be doing is

Integrate[r BesselJ[m, BesselJZero[m, n] r/a]^2, {r, 0, a}, 
 Assumptions -> n \[Element] Integers && Positive[n] &&
                m \[Element] Integers && Negative[m]]

1/2 a^2 (BesselJ[m, BesselJZero[m, n]]^2 + BesselJ[1 + m, BesselJZero[m, n]]^2 -
(2 m BesselJ[m, BesselJZero[m, n]] BesselJ[1 + m, BesselJZero[m, n]])/BesselJZero[m, n])

Sometimes, people forget that Positive[]/Negative[]/NonNegative[]/NonPositive[] are usable as assumptions.

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I've updated the question with some further questions that arise from this answer. –  Eli Lansey Jul 12 '12 at 14:24
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