Integrals of Certain Functions over
6.3 Improper
(-oo'
oo)
lo=lr+Y2+Yr+
Y4
Figure 6.6 Ciosed contour for Example 3'
again approaching zero as p
-
->
oo since a
As a result, on taking the limit
naz
f
lim Lt-.ii'*lrrI+ez
=
az
>
0'
as
p --) co we have
:
(1
/
\ -
-,r
ra2ni
/)
p.u.
f@
edx
/ -JJ-cor+e"
dx
'
(8)
(8). For each
Now we use residue theory to evaluate the contour integral in Eq.
pole
function eo,
+ e2) is analytic inside and on l, except for a simple
p > O,the
l(I
al z : rt i, the residue there being given b
s(zi)
-
|
-az
j2rt + n')1,:,,
":l
"-=i: -"oni
: "oni
(recallExample2,Sec'6,1).Consequently,puttingEqs'(8)and(9)togetherweobtain
o"
./*
#0,: i,n'eni)(-.u'')
-2ni
- ,ati
"-atti
1T
: rtnon' r
EXERCISES 6.3
Verify the integral formulas in Problems
/n
oo
dx
r. P,v./__ 7|Tr;Tt:
2. p.v.
roo
y2
n
rv
dx = )
l ;-b
l-.,o
1xz a91'
l-7
with the aid of residues'
(9)
Residue TheorY
326
fi
f*x2+t
3. I ---;i:
Jo x*+t
={L
,lv
1T
^co
4.pvl_reAlA;T-e
I
\'-oo \
^,/
J
5. p.v.
-
x
f@
s|J-a 1 "-,;.;;P*^
*r4x'l tt)
2'7
\xL
fCO
X'
6.
I (x2+t)lxl+4)
-r-+r-;"'Jo
dx =
rco y6
Zn
rv
zo
J1
I ;-;
.3J'= -GJ0
lx" + lJ
1.
8.Showthatiff(z)-_P(z)lQ(z)isthequotientoftwopolynomialssuchthat
(') dx equals
real z"ros' then p'v' I ?* f
p,
deeQ z 2 + deg *nZ'J'a;;;"
(z) at the poles inthe
-2ni 'llresidues of 'f
9.
Show that
lowet half-plane]
foo o2*
I
P'u,/__ co-sn(z.x) "'"
by integrating
,22
f
with vertices at
cosh(nd around rectangles
z
=
'
Lp' p I i'
-p+t'
10. Given that
integrate
l,
t
e-"
Ol andlet
arovndareclanglewithvertices
P
-->
alz =
O'
p' p * trl'
and )"i (with
@ to derive
@ dx
J, 7 +L
11. Show that
f
zttJ3
sector
by integrating t l(23
io-,-1-r-= rrrt , o
the circular
* r) 1;r1{trr-e o"l"^Ur*tandofletting
p -+ oo'
=b =zft
l3'0 5 r = nJ
discussed in Prob' 18' Exercises
L2. Confirm the values of the integrals
4'7
'
6'3
Improper Integrars of certain
Functions over (-oo, co)
JL /
13. Show thar
[* _]
/-- p];t;i
Summqtion
of
.,. :
d'x
tr(2n)l
;22,',:nQrN, for tt :0,7,2, . ...
Series
14, Let
f (z) be a rational function of the form. p (z) / ek),where deg
e > 2i deg p.
Assume that no poles of (z)
occutat the rnteger. points z : 0, tl, t2,
f
., .. Com_
plete each of the following
st,eps to
the summation formula
".tuilirf,
N
,IT*oI- f (k):
-{sumof theresiduesof nf (z)cot(nz)arthepoles of (z)}.
f
(10)
(a) Show that for the function
I k)
Res(g;
::
r
J.
k)
cot(n z), we have
k): f (k), k:0,!7,!2,.,..
(c) Prove that
rt+ t
N_++oo,.hrv
^,
r
f e)
cot(n z)
dz
: 0,
where I-iy is defined pr.eviously.
(d)
Use the residue theorern and parts (a)
and (c) to derive (10),
15. Using the summation for.mula in prob.
oo1
{a) )
_-
" o?*F n
F
-=--
(b)
(t)
oo1
:
1
t
(n
')
- +)-
14 verify that
n coth(n) IHINT: Take f (7) r/(22 +
=
1).1
--,
-'v
*2
r p : ;
IHINT: The formula in prob, 14 needs ro be modified
ro com-
pensate for the pole of
16. Show that for n a positive
f
(z)
integer.,
= 1 f z2 at z :
0.1
Residue TheorY
328
wheretheconstants82,arctheBernoullinumbers,whicharedeflnedbythepower
n z cotQr
z)=
it-
rro
k=0
17. Show that if a is real and noninteger
and
0<
ffi.rrn
rr'0 .,
r < 1'
oo1
(a)
i --.=l--"
-- n2 csc2 Ta
' r-11*
lk + q'
oo177
(b)
t =; __r:-cotnta
' P-A*k'+a'
u
oo (1, - r\2 - a2 n2 I --=--_----i
- cos2nr
(e) t ,:-=-=
2 (sin2 nr + sirthz tta)'
1:-co l(K - ry a oz12
preceding identities valid?
coshZna
(f) For
which compl' uuiu"' of
iLt'/r f-llof tk) = -{sumof
a
arcthe
residues of
nf (z)csc(ne) atthepolesof
/}'
k=-o
verify that
1,9. Use the formula of Prob' 18 to
6 I
s- \-
rrk
r,,
2k2
6,4
*2
12
ImProPer Integrals Involving
Trigonometric Functions
ourpurposeinthissectionistouseresiduetheorytoevaluateintegralsofthegeneral
forms
P(x)sfrftLx
16 D(y\
,. ..,.,-.
ax',
p.v. |f*
dx'
mx
tot
*
'
P'v'
=r /
J-*
Q@)
l-a
9\x)
Residue TheorY
is
respnlyhavetoperform^oneevaluationofresidues,intheuppel
pu't' ut the end' However' this shortcut
half
not
"h
o' imaglnary
since
p,".f zYarrReP'v I:#'.
|
J-6e
"
is pure imaginary'
In fact the left-hand member
as
we have seen'
EXERCISES 6.4
Using the method' o! resid'ues'
1.
p.v,
€
f
cos(2'r) ,-o,
J_*Z *,
in Problems I -3 '
verifl the integral formulas
=
ft
a
- .r sinx a* :
J_*p_z*to-"
X
foo
3. I ;;::e d*: r;
2. p.v'
f
sin 1)
Tl
COS
r0 v'+
2ecos 1 *
3e3'
L)
in Problems 4-9'
Compute each of the integrals
4.p,v
f@
o3ix
| ;-or
J-(A'
5. p.v.
f
@
'r sin(3x)
,-
J_*_F y + "^
ta o-2ix
dx
6. p.v'
L J-6^
r ""+4 '
I +-
7. pv'
f*
d,
"o"'J_*g*lgTq-^
f@ x3 sirr(Zx) ,-.
8. | 1-; ,-;Q "^
r0 \x" f r/
e. p'v'
r@ cos(2x)
J_*;:u
o*
10, Derive the formula
nn
/-
91-d.x=
\Ti;,-,,
if Imu > 0,
if Imu < 0'
6,5 Indented
Contours
11. Give conditions
331
uncler which the following for.mula is valid:
P.v.
= 2r i
,
lca
P/r.\
etnix :-:://
J -c.c
Ulx )
p
[."riou", of eimz k) I ek)
I
dx
at poles
in rhe upper half_plane]
,
12. Given that
ff ,-r' ,!* : f tt /2,integrate eir, around,the boundary of the circular
sectorSo :
[z = reie : 0 < d < 7r/4,0 S, S O], andt"tp-r-+* toprovethat
/@
J,
6.5
/:-
,l*: ffU
,ir'
*11.
Indented Contours
were assumed to be defined and continuous
now to the probiem of evaluating special
celtain finite points, Our flrst
rmprot)er integratsor
/
;i:i:,'lit,,1lL'ri
over the inrervars
fc
l"
.ta
oi^Zoiol,"off^
to
,n"
tiry ['-' y6;ar,
tfrtctx::r-->l)r
Jn
1b
I
,t.p i,
I trl clx ::
rb
tiry*
J,_,
1{*) a * ,
and
fb .(x\dx
J, f
':,[T.
['-,
f (x)ctx*,1T.
/"
provided the appr.opriate limit(s)
exists, For example,
11 1
l" io':
J0
\/x
fI t
s-+0+,/, +:
tr
liry l'
in
1b
J,"*,Irrtnr,
11
z'Gl
: ;;t j- _\*,:
.f+[)r
and therefole one can say that the
area uncler the graph in Fig, 6.9
vertical asymptote.
of
(r)
r,
is
Jinite despite the
on the other hand, the areas on either side
of the verticar asymptote in the graph
f (x) : 7 / (x 2), depicted in Fig, 6.
-
1
0, are borh infinire, because
Resiclue TheorY
in Fig. 6'15' Then since
and indent around each of its simple poles, as indicated
obtain
is analytic inside the closed contour, we
/(z)
- fP \ ,rzix
"+ Ifr-rz
(/ l'r-r-,r
+l l+
\\/ |-p
J-r+',' l*',f xl -1"d.x*l,r*J,.,IJp:O,
where
fr,
J,.r,
arctheintegrals of
J,
f
(z) over
St',
S'2'
'
Ct'
(9)
respectively' Now by
Jordan's lemma we have
fg/p:o'
and from Eq. (5) of Lemma 4,
lim
r1-+Q*
J,.,L
- -in
Res(-1) =
lim.(z + 1)/(z)
-in z+-L
ai-
Z€''"'
-in z.->-lZ- |
lim
:-!?,L
-.,;
and
l\m J,, : -itt Res(l) : -in
]1qr
z-->
r2-->Q! '
(t - l) f
.;,
Ze''
'"::s
"
(z)
,
)i
e-'
-tr
r+1:
2
Hence on taking the limits in Eq' (9) we get
p".
rrx *"2ix ,
/ ;2-1dxJ-o L- L
ine2i
ine-2i
n
_.,- ^^-
r
--+;-o=izcos2'
EXERCISES 6.5
1.
given circular arcs'
Compute each of the foliowing limits along the
t o-Zt
(a) lim,.*e+
,
(b) lim,.-s+
|
o3iz
I z'- t
Jl,
-dz,wherclv
7T
jr
i
,"
7': llre'o' 4 3a < r
:
dz,whercTr i z
JT, u
f t--roZ
-1 dz,where S'' : z
lim,.-s+
f ?y
(d)
.^
I +dz,whereTv"7:re'a'0=e =t
JT' '
l"
:
! + re-io,n :. o < Ztr
re-iq
'1T
< 0 < 2n
2.8,
(Jsing the technique of resid,wes, l,erify each of the integral formulas in Problems
2.p.v,
rco
I
o2ix
|
=dx=rTie-"'
'J-Cn ^
'
6,6 Integrals
3.pv
Involving
M"ltiplqvulltd
I:6+
-- (x
- ,)
1)(x
34s
F.onttignt
)- - /r-; ("' - n)
-
uA
L
)
,{fco
J,
6. p,v.
7. p.v,
lx'
sinx
J-*Tr, +q
f@ ;r cos.x
I -J^
J-6^
8. p.v.
G _ r)0,
dx
|
:
:
7t
T
n [sin(1)
10. Verify
-
2sin(2)]
,,' + V3cos(l)lr + z sin(2)
fsin(l)
,./
9. Computep.v.
Lcos(r.)
L
7r -.nf
/m cos(2x)
a-*:1t-v:
/-oo ;i*
,
^
|- ,,, c'-21
- I
3
(3r'' ,3ix !\ I
- -tl
.. T
t'lr'r".
1ru -:..3
/ ^
J-oo
4
lHrNr'sin3x=0"(?
\ L
that
I
"/ )
x dx:
--'
J'
r_
sinz
7T
: | {t - 2D : lRe (1 - e2i")']
"ot
16 edx
11, Compute P.v. /
1 rlxfor-0 <a <1"IHINTI
/ _ca .
alound the Points z - 0 and z : 2n i'l
IHINT: sin2 x
Indentthecontourof Fig'6'6
,
12. Verifythatfora > 0 and b >
0
-,1,
-'
Jo ;P;$u^ - 2b'\'
f*
6.6
sin(ax)
: !: (t
n-rtb\.
/
Integrals Involving
MultiPle'Valued Functions
Inatternptingtoapplyr.esiduetheorytocomputeanintegralof/(x),itmaytufnout
If this happens, we need to modify
that the compiex r"".ti"" ii.l i, *,1i,lpr"-ua^1ued.
singularities but also bt'anch
our procedure by takrng into account not only isolated
to integrate along a branch
ooints and brancl cuts, I' fact we may flnd ii ne"ersary
cut, so we turn fir'st to a discussion of this technique'
not an integer, and let ./(z) be the
To be specific, let a denote a real number, but
''
of z to lie between 0 and 2n that ts'
branch of zo obtained by restricting the argutnent
J'Q):'u(Logrlil)' wheLez':rei1'0<0 <2n'
(1)