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IntroductionExtensive tables of the Bessel Functions Yo(x),Yi(z),Ko(x),and Ki(x)are included in Bessel Func-tions,Part I,Functions of Orders Zero and Unity,vol.VI of the series of tables published by the Mathe-tical Tables Committee of the British Association for the Advancement of Science.This volumewill be referred to hereafter as the BAAS table.Because of the difficulty of interpolation in the tablesof these functions in the BAAS volume for all arguments (due to the logarithmic singularity at theorigin),certain auxiliary functions to be defined below were tabulated in the BAAS table in the regionbetween 0 and 0.5.Because of the frequent need for values of the Bessel Functions Fo(x),.Yi(z),Ko(x),and Ki(x)forall arguments in ny physical and engineering problems,the Computation Laboratory of theNational Bureau of Standards has computed these functions in the region between 0 and 1 at a con-siderably finer interval than in the BAAs tables.The resulting tables presented in this issue of theBureau's Applied Mathetics Series will enable the user to obtain almost the full accuracy of thetable,over most of its range,by linear interpolation.Specifically,these tables give the values of Yo(z)and Yi(z)with first and second differences for z=0(.0001)0.05(.001)1 and the values of Ko(x)and Ki(x)with first and second differences for z=0(.0001)0.033(.001)1.To simplify interpolation in the present table of Y(z)and Yi(x)in the all region between 0 and0.0050,the same auxiliary functions given in the BAAS table have been tabulated here at an interval of0.0001.Similarly,auxiliary functions related to Ko(z)and Ki(x),the same as in the BAAS tables,havebeen tabulated for z=0(.001).030 for the sake of ease of interpolation in the last mentioned functions.With the exception of a few entries close to the origin,the entries of the present table were obtainedby interpolation in the BAAS tables and they are given to the same number of significant figures.Thelast place y therefore be in error by about a unit and in some instances perhaps by two units.Aall number of entries near the origin,which are given to eight or nine significant figures,were com-puted independently;the ninth significant figure,where given,y be in error by as much as four units.The Functions Y(x)and Y(x)The general solution of the Bessel differential equationdxy=AJ,(x)+BY,(x)whereY(z)-COe(vx)J.(z)-J--(z)and where A and B are arbitrary constants.When y is an integer n,J(x)=(-1)"J(x).In this cas(1sin yrwhich,for n=0 and n=1,yieldswhere y is Euler's constant 0.57721 56649 01533 ...For purposes of interpolation in the neighborhood of the origin,it is convenient to tabulate thefunctionsIn terms of the above functions,we haveThe functions Ye(z)and Yi(x)satisfy the Wronskian relationand the relationsThe Functions K(x)and Ki(x)The general solution of the modified Bessel differential equationiswhereand where A and B are arbitrary constants.When r is an integer n,I(z)=I().In this case,(2)which,for n=0 and n=1,yieldsFor purposes of interpolation in the neighborhood of the origin,it is convenient to tabulate thefunctionsEo(x)=Ko()+I(=)log.z;E(z)=Ki(z)-I()log.
