114 Mr. 0. Heaviside. [June 15, 



provided n is not a negative integer, when, we know that closer ex- 

 amination is required. 



Apparent Failure of Numerical Equivalence in certain Cases. 



34. Now, although the equations following (35), (36) (excepting 

 (43)) are deducible from them by the process used immediately and 

 without trouble, there is considerable difficulty in finding out their 

 meaning. Considering (37) and (38), I knew that in the case 

 n =. \ the equivalence was satisfactory all round, though not very 

 understandable. When n is 0, or integral, it is also satisfactory, for 

 then we have merely a perversion of terms in passing from u to v. 

 But when I tried the case n = ^, and subjected it to numerical 

 calculation, with the expectation of finding numerical equivalence to 

 the extent permitted by the initial convergence of ' the divergent 

 series, I found a glaring discrepancy between u and v. Furthermore, 

 on taking n = 1, we produce 



u = e-*, (44) 



\2 |3 



(45) 



which show no sort of numerical equivalence whatever. Similarly, 

 n = 2 gives 



u = l-2aj+ x z -x*+ . , (46) 



.r- 2 / 2 3 2 2 3 2 2 2 3 2 4 a 



which also do not show any numerical equivalence. I was therefore 

 led to think that the equivalence in the case of the Fourier. Bessel 

 function was due to some peculiarity of that function, and it is a fact 

 that the function is the meeting-place of many remarkabilities. The 

 matter was therefore put on one side for the time. But, more 

 recently, independent evidence in other directions showed me that 

 there was no particular reason to expect such a complete failure. 

 And, in fact, on returning to the discrepant calculations relating to 

 n j, I found an important numerical error. When corrected, 

 the results for u and v agreed as fairly as could be expected. 



