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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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3. Auxiliary Functions and Arguments
One of the objects of this Handbook is to pro-
vide tables or computing methods which enable
the user to evaluate the tabulated functions over
complele ranges of real values of their parameters.
In order to achieve this object, frequent use has
been made of auxiliary functions to remove the
infinite part of the original functions a t their
singularities, and auxiliary arguments to co e with
The exponential integral of positive apgument
is given by
infinite ranges. An example will make t % e pro-
x 22 xa =7+lnx+-+-+-+ . . . 1.1! 2:2! 3.3!
The logarithmic singularity precludes direct inter-
polation near z=O. The functions Ei(x)---In z
and cc-’[Ei(z)-ln 2-71, however, are well-
behaved and readily interpolable in this region.
Either will do as an auxiliary function; the latter
was in fact selected as it yields slightly higher
accuracy when Ei(z) is recovered. The function
P[Ei(z)-ln 2-71 has been tabulated to nine
decimals for the range O<x<$. For 3 5 2 5 2 ,
Ei(2) is sufficiently well-behaved to admit direct
tabulation, but for larger values of 2, its expo-
nential character predominates. A smoother and
more readily interpolab€e function for large 2 is
ze-$Ei(z); this has been tabulated for 2 52 510.
Finally, the range 10 5x50~ is covered by use of
the inverse argument z-l. Twenty-one entries of
ze-”Ei(x), corresponding to ~-~=.1(-.005)0, suf-
fice to produce an interpolable table.
The tables in this Handbook are not provided
with differences or other aids to interpolation, be-
cause it was felt that the space they require could
be better employed by the tabulation of additional
functions. Admittedly aids could have been given
without consuming extra space by increasing the
intervals of tabdation, but this would have con-
flicted with the requirement that linear interpola-
tion is accurate to four or five figures:
For a plications in which linear interpolation
is insdciently accurate it is ’intended that
Lagrange’s formula or Aitken’s method of itera-
tive linear interpolation3 be used. To help the
user, there is a statement at the foot of most tables
of the maximum error in a linear inte olate,
Lagrange’s formula or Aitken’s method to inter-
polate to full tabular accuracy.
As an example, consider the following extract
from Table 5.1.
and the number of function values nee 3 ed in
X xezEl(x) X xe*El (x)
7.5 .89268 7854 8.0 .89823 7113
7.6 .89384 6312 8. 1 .89927 7888
7. 7 .89497 9666 8. 2 .90029 7306
7. 8 . 89608 8737 8. 3 .90129 6073
7. 9 . 89717 4302 8.4 .go227 4695 “-”I
The numbers in the square brackets mean that
the maximum error in a linear interpolate is
3 X and that to interpolate to the full tabular
accuracy five points must be used in Lagrange’s
and Aitken’s methods.
* A. 0 Aitkea On inte olation b iteration of roportional parts with-
out the use of diherences, %c. Edingurgh Math. 8oc. 3,66-76 (l932j.
Let us suppose that we wish to compute the
value of ze”El(z) for 2=7.9527 from this table.
We describe in turn the application of the methods
of linear interpolation, Lagrange and Aitken, and
of alternative methods based on differences and
(1) Linear interpolation. The formula for this
process is given by
j P = (l--P)jO+Pfl
where jo, jl are consecutive tabular values of the
function, corresponding to arguments xo, zl, re-
spectively; p is the given fraction of the argument
and jp the required interpolate.
instance, we have
P = (-zo)/(21 -20)
In the present
fo=.89717 4302 fiz.89823 7113 p=.527
The most convenient way to evaluate the formula
on a desk calculating machine is to set jo and ji
in turn on the keyboard, and carry out the multi-
plications by 1-p and p cumulatively; a partial
check is then provided by the multiplier dial
reading unity. We obtain
f.ag.r= (1 -.527) (39717 4302) +.527(.89823 7113)
= 39773 4403.
Since it is known that there.is a possible error
of 3 x in the linear formula, we round off this
result to 39773. The maximum possible error in
this answer is composed of the error committed
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