ECH 140 Study Guide  Joule, Friedrich Bessel, List Of Recurring Futurama Characters
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Course
ECH 140
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SEC. 5.5 Besse['s Equation. Bessel Functions/"(x)
(1)
(2) \r
y(x)
: ) ax""
nt:o
(a6 * 0)
(a)
(3) (b)
(c)
r(r  l)as
* ras
 v2ao: g
(r l l)ra, + (r + l)ar  vzar: g
(.s
+ r)(s
* r l)a" * (s * r)a" * arr v2a": g
:o\
 l\
').
From (3a)
we obtain the indicial equation by dropping a6,
(r+v)(rv):0.
The roots are rt: / (> 0) and
rr: ,
"FRIEDRICH
WILHELM BESSEL
(17841846),
German astronomer and mathematicran. studied
astronomy
on his own in his spare time as an apprentice of a trade company
and finally became director of the new Kdnigsberg
Observatory.
Formulas on Bessel functions are contained in Ref. [GRll and the standard treatise
[Al3].
(s
(.r
(s
(4)
I
)
$,il
189
5.5
Bessel's Equation.
Bessel Functions
J,(x)
One of the most important ODEs
in applied
mathematics in Bessel's equation,6
*'y" + xy'
+ 1x2
 v2)y: o.
Its diverse applications
range from electric fields to heat conduction and
vibrations
(see
Sec. I 2.9).
It often appears
when a
problem
shows
cylindrical
symmetry
(ust as Legendre's
equation may appear in cases of spherical symmetry).
The parameter
zin (1) is a given
number.
We assume that z is real and nonnegative.
Bessel's equation
can be solved by the Frobenius method, as we mentioned at the
beginning
of the preceding
section, where the equation
is written in standard form
(obtained
by dividing (1) by x2). Accordingly, we substitute the series
with
undetermined
coefficients and its derivatives into
(l). This
gives
) r* * r)(m
* r  l)a*x*' * ! 1m
t r')a*f"'
m:O m:O
+) ot+r+2  ,') or^*'  0.
m:O m:O
We equate the sum of the coefficients of x"*' to zero. Note that this power r"*'
corresponds to m : s in the first, second, and fourth series, and to m : s  2 in the
third series. Hence for s : 0 and s : l, the third series does
not contribute
since
m> 0.
For.s
: 2,3,' ' ' all four series contribute, so that we
get
a
general
formula for
all these s. We find
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r90 CHAP. 5 Series
Solutions of ODEs.
Soecial Functions
CoefficientRecursionforr
= rr= t/. Forr  4Eq. (3b)reduces
t<t(2ui l)u1
Hcnce
rrr
: 0 since v > 0. Substituting r : u in (3c)
and
conrbining the three
contuinirtq rr"
gir
cs sirnply
(5) (.r'* 2z).ra.,
I u.
2 0.
Since ar : 0 and
y > 0. it tbllows from
(5)
that rrr
: 0. c,
: g,
to deal
only with eyenruunbered
cocfficients
rr"
with
s
 2nt. For .r'
:
(2nt f 2u)2n102r,,
* tt2,,,.2: 0.
givcs the lecursion lbrrnula
' ' . Hence we have
2trt.
Et1.
(,5)
becomes
Solving ftl'rr2,,,
(6) I
Lt2n' a) u2',, 2'
lDt\v t nt)
Frorn
(6)
we can now
determine
d2, e4,. . . succcssively.
This
gives
(I
(l
LI2
(lz
22(u
+ l)
(7)
(l^
and so
on, and in general
Integer
values ofv are denoted by n.Thi
(9) 0o
because
then rrl(ri
+ l) ' ' . (n I tn\

(
l0)
222(v
+ 2) 2421
(v + l)(z + 2)
(
 I)"'n6
02r,t 22'"m,.
(u+ l)(z + 2). . . \u t m)
Integer
z :n
 rr
the
relation
(7)
becomes
is standard.
For r
( I
)"'110
t" 22"ttnt 1ri  l)1rr I l.t . . ttr I tttl
ao
is still
arbitrary. so
that the
series
(2)
with
these coe fficients
would contain this
arbitrary
fiictor
rtn. This would be a highly impractical
situation fbr developing forntulas
or
computint values of this new function.
Accordingly. we have
to ntakc a choice.
ao
= 1
would be
possible,
but more
practical
tunts out to be
I
2" n1.
(nt * n)t in (8). so
that
(8) simply becontes
( I
)"'
(lzr,,
22"' " tnt (n i m\1.
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. 5.5 Bessel's Equation. Bessel Functions/,,(x)
This simplicity of the
coefTicients and 11
:
and
given
by
(l
l)
EXAMPLE I
denominator of (10)
partially
motivates
the choice
(9).
With these
u
 n we
get
from
(2)
a
particular
solution of (l
).
denoted
by l"(x)
l9l
(
 7)^x2"'
22"tnmt (.n * m)t '
J,,(,r)
is
called the Bessel function of the first kind oJ ortler n. The series
(11)
converges
fbr
all ,r. as the ratio test shows. In fact, it
converges very rapidly because of the tactorials
in the denominator.
Bessel Functions,fo(x)
and,f,(x)
For ir  0 ue obtain ll'onr
(lI.t the Bcssel function of order 0
./"(r)
: r'i
m:o t,t
/t'cy, 
lt2)
113) Jr(o:
i
,r .0
 r I Jil,2ilr
,,..
_
\s
Jo\^ t  zJ ,2tt, ,.2
,,, o ! \nl:l
246
I{I
_+
220,.2

I wc obtain the Bessel function of order I
357
J.TY ._
t" l t 2: 2''2:
1: t'1l
,ll
which looks simillr to a cosine
(Fig.
107). For ri 
( 11""2"" I
2znL+rmt (in + l)l
I
2
u'hich
looks siruiltr to a sine
(Fig.
107). But
the
zeros
of thcse tunctions are not completely regularly
spaced
(see
also Tahle Al in App.
.5)
ancl
the
height
of thc
"waves"
decreases with increasing.r. Heuristicirlly,
n2l.tz
in (

) in standard
firrm
[(
l) dividcd by .r'l is zero
(il'r  Q)
ur small
in absolutc
value
tbr large ,r. und so is
1'l.1. so that then Bessel's
eqrration conrcs closc tu r'" + r'  0, the equatiun of cns
r
and
sin
.r: also r''l.r acts
as a "tlampint
tem." in part
responsible lbr the decrease
in hcight. C)ne cun show that ftrr large,r.
(
l4) r,,(,)
8,.",(.,
T

where
 is read
"asymptoticalll' equal" and mcans thatJor.fi.red r?
the
quotient
ot the two sidcs approaches I
AS,f + Z.
Formula
(1,1)
is surprisingiy
accurate even lirr smaller r (> 0).
For instancc.
it wlll give you good
starting
values in a computer
progfarn
1br the basic
trsk of computing zeros. For example. lirr the first
three
zeros of J6
vou
obtain the
vahres
2.356
(2..105
exact to 3 decimals. error 0.049). 5.'19U
(5.520.
error
0.022). 8.639
(8.65.1.
crror 0.0I
5
).
ctc
;)
T
,); 
Fig. 107. Bessel functions
of the first kind
Jo
and
J,
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