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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 32 "Bessel Functions J0(x) an
d J1(x)" }}{PARA 256 "" 0 "" {TEXT -1 18 "Original Version: " }{TEXT 
265 15 "Earl C. Swallow" }{TEXT -1 34 " 12/8/98 for PHY-411 Modern Opt
ics" }}{PARA 257 "" 0 "" {TEXT -1 48 "Revised for more general audienc
e: ECS  10/10/99" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 189 "Bessel functions occur frequently in applied mathematics
, physics, chemistry, and engineering.  Problems of many kinds (in ato
mic and molecular theory, optics, electromagnetics, mechanics, " }
{TEXT 266 4 "etc." }{TEXT -1 194 ") with cylindrical symmetry give ris
e to Bessel functions.  Here we take a brief look at the first two int
eger order Bessel Functions of the first kind, commonly designated by \+
J0(x) and J1(x).  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 111 "Properties of Bessel Functions and tables of numerical v
alues were traditionally found in math tables like the " }{TEXT 270 
24 "CRC Standard Math Tables" }{TEXT -1 46 ".  Maple provides them as \+
standard functions: " }{TEXT 268 20 "J0(x) = BesselJ(0,x)" }{TEXT -1 
5 " and " }{TEXT 269 20 "J1(x) = BesselJ(1,x)" }{TEXT -1 3 ".  " }}
{PARA 0 "" 0 "" {TEXT -1 112 "(I bet you can figure out the designatio
n for Bessel Functions of other orders after seeing these two examples
.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"assume(x, real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "First I just look at the values at
 the origin to check syntax, " }{TEXT 267 4 "etc." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "BesselJ(0,0), BesselJ(1,0);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 67 "Looks good -- the same as in math tables.
  Let's plot.  Start with " }{TEXT 258 5 "J0(x)" }{TEXT -1 7 ", then \+
" }{TEXT 259 5 "J1(x)" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "plot(BesselJ(0,x), x=-20..20);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 3 "So " }{TEXT 256 5 "J0(x)" }{TEXT -1 26 " is an even f
unction like " }{TEXT 257 6 "cos(x)" }{TEXT -1 106 ", except that it h
as decreasing oscillation amplitude and the zeros don't appear to be e
venly spaced like." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(Bessel
J(1,x), x=-20..20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "And " }
{TEXT 261 5 "J1(x)" }{TEXT -1 13 " is odd like " }{TEXT 262 6 "sin(x)
" }{TEXT -1 71 ", but again with decreasing amplitude and unevely spac
ed zeros.  Also, " }{TEXT 273 5 "J1(x)" }{TEXT -1 24 " doesn't reach +
1 or -1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Now let's directly compare
 " }{TEXT 271 5 "J1(x)" }{TEXT -1 5 " and " }{TEXT 272 6 "sin(x)" }
{TEXT -1 27 " by plotting them together." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "plot([BesselJ(1,x), sin(x)], x=-20..20);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 46 "For a bit better comparison, we normalize
 the " }{TEXT 260 6 "sin(x)" }{TEXT -1 49 " plot so the greatest maxim
um and minimum match. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot([BesselJ(1,x), 0.56*
sin(x)], x=-20..20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "We can al
so \"tweak\" the location of the first zero to match the cycles better
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "bz1 := solve(BesselJ(1
,x) = 0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "bz1 := evalf
(bz1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "scale := Pi/bz1;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot([BesselJ(1,x), 0.5
6*sin(scale*x)], x=-20..20);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "W
e see that the oscillations of " }{TEXT 274 5 "J1(x)" }{TEXT -1 22 " b
ecome more rapid as " }{TEXT 275 1 "x" }{TEXT -1 121 " increases.  It \+
would be natural to do J0(x) and cos(x) comparison next.  I leave this
 as an \"exercise for the student.\" " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 70 "Now get numerical values of some zero
s of J1(x) for reference.  Using " }{TEXT 276 6 "fsolve" }{TEXT -1 36 
" does this more directly than above." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "bz1 := fsolve(BesselJ(1,x) = 0, x, 0..4);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 197 "      I know there is an automatic facil
ity to get zeros in Release 5 of the full version of Maple (as in SC02
5).  Problem SOLVED -- I was getting ranges from graph with scale*x on
 horizontal axis!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "bz2 :=
 fsolve(BesselJ(1,x) = 0, x = 6..8);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "bz3 := fsolve(BesselJ(1,x) = 0, x = 8..12);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "bz4 := fsolve(BesselJ(1,x) =
 0, x = 12..15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "bz5 := \+
fsolve(BesselJ(1,x) = 0, x = 15..18);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "bz6 := fsolve(BesselJ(1,x) = 0, x = 17..22);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "These match the values given in th
e " }{TEXT 277 23 "CRC Standard Math Table" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "The full version
 of Maple V, Release 5 (as in SC025) has an automatic facility for obt
aining zeros of Bessel Functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 43 "Below I look at some patterns of the zero
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "evalf(bz1/(1.22*Pi))
,evalf(bz2/(2.22*Pi)),evalf(bz3/(3.22*Pi)),evalf(bz4/(4.22*Pi)),evalf(
bz5/(5.22*Pi)),evalf(bz6/(6.22*Pi));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "bz2-bz1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
bz3-bz2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "bz4-bz3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "bz5-bz4;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "bz6-bz5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
152 "It looks like the spacing starts out slightly greater than Pi and
 is decreasing toward Pi as we go outward.  This could be checked by g
oing out further." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Now look at the se
ries expansion of " }{TEXT 263 5 "J0(x)" }{TEXT -1 5 " and " }{TEXT 
264 5 "J1(x)" }{TEXT -1 9 " for fun." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "series(BesselJ(0,x),x,10);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 26 "series(BesselJ(1,x),x,10);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 15 "Thinking about " }{TEXT 278 5 "J1(x)" }{TEXT -1 16 "
 in relation to " }{TEXT 279 6 "sin(x)" }{TEXT -1 18 ", we remember th
e " }{TEXT 280 8 "sin(x)/x" }{TEXT -1 17 " approaches 1 as " }{TEXT 
281 1 "x" }{TEXT -1 45 " approaches zero.  Looking at the series for \+
" }{TEXT 282 5 "J1(x)" }{TEXT -1 16 " above, clearly " }{TEXT 283 7 "J
1(x)/x" }{TEXT -1 19 " approaches 1/2 as " }{TEXT 284 1 "x" }{TEXT -1 
54 " approaches 0.  Let's ask Maple to evaluate the limit." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit(BesselJ(1,x)/x, x = 0);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Let's see what happens if we simpl
y substitute x  = 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sub
s(x = 0, BesselJ(1,x)/x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "algsubs(x = 0, BesselJ(1,x)/x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 34 "UGH!! The second one gives us the " }{TEXT 285 5 "WRONG" }
{TEXT -1 108 " answer without further comment.  You should always be s
keptical about subtle answers from computer programs" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "R
emembering that " }{TEXT 286 8 "sin(x)/x" }{TEXT -1 66 " describes the
 diffraction pattern produced by a narrow slit, and " }{TEXT 287 7 "J1
(x)/x" }{TEXT -1 109 " describes the diffraction pattern produced by a
 round aperture, I compare the two by plotting them together." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 74 "plot([(2*BesselJ(1,x)/x), (sin(x)/x)], x = -20..20,
 color = [red, green]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 216 "Clearly we could go forwar
d by squaring these distributions to get the light intensities in the \+
diffraction patterns, and then find the locations of the diffraction z
eros (dark fringes) and maxima (bright fringes)..." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 32 }
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