Rit (Vísindafélag Íslendinga) - 01.06.1951, Blaðsíða 16

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x— a" / \ dnv(x) xan
2_A,/«#qp(i1)-^íyB—A5
n — o '
=^(elnx+0,Dí}) -»qp(n)=e<flö’?Ðinx-» j^(x)cp(n) (5)
xr— rf"«p(ri) dny(x) con ,
í/n" </(!nx)" ' n! 6
=9?(n + ®Ðlnx) -» (6)
=^(ln[e’J(l + Ain*/<1,)])-»^(Jc) B
Above 1 have entered some of the series which we can
readily derive from the Basal Function A, indeed their de-
rivation is so obvious that we can write them down strait-
forwardly. In this list I have included only those infinite
series which I consider most important. They are marked
Ai to A6 their respective symbolic expressions are also in-
serted.
As to the denotations we must keep in mind that
v (jc)
means that y(x) is to be differentiated n times with
regard to x“ and afterwards we put xa —o.
dny(x),
d(lU)
we must, of course, put x=oo after n differentiations with
regard to V*.
All the series At to Ac are, according to my view, equi-
valent. They are, however, not equally well fit for numeri-
cal calculation in special cases. It will be our task in each
case to select those which best serve our purposes.
The series AL—A6 may, of course, be derived in usual
way by means of Taylor’s theorem. However, the symbolic
derivation, as indicated above, is so simple that it deserves
due attention.
As the symbolic operators av and Ðv cannot generally
be treated as algebraic quantities if they are mixed up with
-ri in the formulae, we have, as a rule, to exclude ri from