A Simpler F o r m u l a for Certain Integrals
Ur '
in t h e T h e o r y of t h e M ö s s b a u e r L i n e S h a p e
i y-m I'
dz
(z + iy)»> (z-iy)*
( ^ + 7)
t o * ) " — J
(4)
BRUCE T . CLEVELAND
t h e n it is a p p a r e n t t h a t
Department of Physics and A s t r o n o m y
Qm = Re(Um).
State University of New Y o r k at Buffalo
Buffalo, N e w Y o r k , U S A
I n the u p p e r half of the c o m p l e x p l a n e , the o n l y
(Z. Naturforsch. 27 a. 370 [1972] ; received 10 January 1972)
gularity of the i n t e g r a n d in
It is shown that the integrals Qm which o c c u r in the p r o b lem of the M ö s s b a u e r thick-absorber line shape can be expressed as the real part of a polynomial of degree m.
In their w o r k o n the M ö s s b a u e r
s h a p e , HEBERLE a n d F R A N C O
integral
which
gives
the
thick-absorber
h a v e s h o w n that
2
fractional
absorption
of
recoil-free radiation c a n b e evaluated b y means of
a t z = i y. A p p l y i n g t h e r e s i d u e t h e o r e m , w e
2 y2m
U m
= ~
f
(m-1).'
-1
(z-x+i)
^ „ i d ^ - !
the
theorem
an
- 1 =I 0l l\
d"
(z—x + i) {z+iy)n + l_
dz"
1
^
dzn~l z-x + i
m=1
Qm(y, x),
I
(1)
The
d e r i v a t i v e s in Dn
relation
Leibniz's
3
'd»-'
J
(z+iy)>
(5)
Dn(z) =
(-s)™
1
T h e derivative c a n b e e v a l u a t e d b y m e a n s of
- 30
~
~ 2
m
have
the
oo
f exp{-r=s/(z2+y2)}
=
sin-
is a p o l e of o r d e r
line
infinite series
1
(4)
can
1
'
1
dr' ( z + i
be obtained
+ 1
j '
b y use of
the
( v a l i d w h e n A; is a p o s i t i v e i n t e g e r a n d c is in-
dependent of
x),
where
Qm(y, x)
y2 \m
dz
~ f Z2 + y2/ l+(z — x)2'
=
(2)
W e e v a l u a t e Dn a t z = i y a n d o b t a i n
•-H
1 = 0 l\
F o r the definitions of the v a r i o u s s y m b o l s , the r e a d e r is
r e f e r r e d to R e f .
T h e i n t e g r a l Qm
has been
as [ s e e E q s . ( 2 . 6 ) a n d ( 1 0 . 1 0 ) of R e f .
Qm{y,x) =
[1(7 + 1)]™
Equation (5) then b e c o m e s
_y(y±l)_
o
z 2 + (7 + 1 ) 8
m
y -1 I y(y+1)MZ__1
_,n
expressed
2]
t \ U
2
+(7 + l)2/
m-l
Um(y,x)
"mly-1f 2 m—2 I
i
f
1
\
Qm t h a t is c o n s i d e r a b l y s i m p l e r t h a n E q u a t i o n
in E q u a t i o n
(z-x)
2
] =Re[i/(z-x +
t h e n w e h a v e t h e redegree
f
(7)
(3).
T h e c a l c u l a t i o n of e b y the series in E q . ( 1 )
i)]
( 2 ) . F o r a n i n t e g r a l o f a r e a l v a r i a b l e it
p a r t . T h u s , if w e d e f i n e t h e c o m p l e x f u n c t i o n
Reprint requests to Dr. B. T . CLEVELAND, Dept. of Physics,
S U N Y at Buffalo, Buffalo, New Y o r k 14 214, U S A .
J. HEBERLE and S. F R A N C O , Z . Naturforsch. 2 3 a. 1 4 3 9
[1968].
(6)
1 -fix
^
,
2
/ ' V fm-l + l'
s
Qm(y,x) = -4— Re j 2 n (
)«"
z
for
is p o s s i b l e to c o m m u t e i n t e g r a t i o n a n d t a k i n g t h e r e a l
1
1 + i x),
2y
m with constant coefficients
W e b e g i n b y using the identity
l/[l-f
m-l m-l+l\(
1
I
1=0
s u l t that Qm i s t h e r e a l p a r t o f a p o l y n o m i a l o f
(3)
It is t h e p u r p o s e o f this n o t e t o p r e s e n t a f o r m u l a
4m
I f w e d e f i n e a = 2 y/(y+
i-1
2 m — l — iyi-1
=
performed considerably
with Equation
faster b y use of E q .
can be
(7)
than
(3).
T h e author is grateful to the Research Foundation of the
State University of N e w Y o r k f o r the award of a Faculty
Research Fellowship.
2
3
S. F R A N C O and J . HEBERLE, Z . Naturforsch. 2 5 a, 134
[1970].
See, f o r instance, H a n d b o o k of Mathematical Functions, ed.
M . A B R A M O W I T Z and I . A . S T E G U N , Dover, New Y o r k 1965,
Eq. (3.3.8).
Unauthenticated
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