TRAVELING-WAVE TUBES 51 



Miscellaneous Formulae Involving /„(x) and Kn(x) 



1. /.^i(Z) - /.+i(Z) = ^I.iZ), K^^AZ) - K,+dZ) = - yKXZ) 



2. h.,{Z) + /,+i(Z) = ll'XZ), K.^,{Z) + A',+i(Z) = - IK'XZ) 



3. Z/;(Z) + j'/.CZ) = Zh,^{Z), ZK'JZ) + ^A'.(Z) = - ZA'._i(Z) 



4. Z/;(Z) - vI^Z) = ZL+iiZ), ZK'XZ) - vK^Z) - - ZA%+i(Z) 



= (-)"'Z'-"'A_(Z) 



/ j/ \[L{Z)\ ^ LUZ) /_^\ 



\Z^Z/ 1 Z- J Z''+'« ' vz</z/ t z 



7. /o(Z) = /i(Z), Ao(Z) = -Ai(Z) 



KXZ)\^ ^_yn K. + ,n(Z) 



8. /_.(Z) = /.(Z), A_.(Z) = A,(Z) 



^1/2 



.2Z, 



9. Ai/2(Z) = ( — ) e- 



10. /.(Ze'"'') = r'"^7.(Z) 



11. A.(Zr"') = e-'-^'K^Z) - i ^J^L^^ IXZ) 



sm vir 



12. /.,(Z) A.+i(Z) + /„+i(Z) A.,(Z) = 1/Z 

 For small values of X : 



13. h{X) = 1 + .25 X- + .015625 X' -^ • • • 



14. h{X) - .5X + .0625 .Y=' + .002604 X' + • • 



15. Ao(X) = -I7+ In 



(|)}u..) + i.v= + Ax' + 



/ 



16. A-.(.Y) = {. + ,„ g)} 7.(.Y) + J^ _ 1 X _ ^ .V» + . . .} 



7 = .5772 . . . (Euler's constant) 

 For large values of A' : 



e"" f, , .125 , .0703125 , .073242 



"■ ^"(■^■' - (2^-- i' + ^ + '-^^ + -^ + 



