hw1_solutions - Phgsics 325 Fall 2006 Solutions to Homework...

Phgsics 325, Fall 2006
Solutions to Homework Assignment
#I
i)
Taylor expansion of erf (r) :
#
f;
e-"u'du. we need the first three nonzero
terms,
evaiuatedatr:0.
erf(0)
: g
erf'
-
rtu:"
by the fundamental
theorem of calculus.
erf,
:
h
u, r
:
0.
erf't =
rtl-Zr"-"]
which
vanishes at c:0,
er.ft"
:
hl-r"-n
+ 4rr"-,"1which has-the
value
ff
at r
:
0.
erfrrrt
-
ftl+re-,'
*8xe-,2
-
Brt"-,'l which
,runi.n",
at r
=
0.
:: {""' ;,k
[-tle-"
+ 4e-" + Be-,''- I6r2e-,'
-
24rze-,'
*
r6rae-,rl which has
r]re
value
#
at r:0.
The firit three nonzero
terms of the Taylor expansion
are
e r f ( r ) = o *
h * o - # + o + m
Hete's an elegant alternative solution (easier,
tool).
you
can express
e-u,
:
Ino
g#f
,
and integrate that ,:l.,T,l{,t.rm
to get a series expansion
for eri(x).
erf(x):rtD|y)ffi
Reading off the first'three
terms gives
e r f ( r ) = * 1 " - 4 + t l l
Which agrbes' with it
. u"U'ou. computation.
Inserting r:
I to the seriesexpansion, erf(I)
= 0.g6b1, which can be compared with
the value looked up in a table (looking up erf(l) in a table is not required for credit),
erf (I) :0.8427.