134 Mr. 0. Heaviside. [June 15, 



and also by 



= log (l + v-') = log ^ = ~ log - , (136) 



which may be useful later. 



Deduction of Second Kind of Bessel Function, K (),/rom the Generalized 

 Formula of the First Kind. 



64. A similar treatment of the generalized formula for the Fourier- 

 Bessel function leads to the companion function. Thus, take 



Io(3) = 2y'[/(r)?, (137) 



as in (76) Part I, the value of y being %x*. Differentiate to r. 

 Then 



= I (oO log y + 2$ 2/'/(r)/'(r). (138) 



Here take the special case r = 0. Then we have 

 = I (a:) logy 



+a{c+y(C-l)+^ r (C-l-J) +^1(0-1-4-*) + .... 



The third line is apparently zero. But it must, as we shall see, 

 be retained, though in a changed form. Or 



(140) 



Another way. Automatic Standardization. 



65. Now the right member is certainly not zero, for it represents 

 the companion of I (#), as may be proved in various ways, classical 

 and nnclassical. One way is from the formula for !(), thus, 



