the residue theorem • Trig and indefinite integrals

!"$#%&'()+*,#-/.1023.04
5"6,()&#7&
8"9:;<>= #():1?,#@,,A9#(B:,?,C,"9D
< 9%()EF+*,:G()*1:,HI*1(BC(B*E+9+JB#
<LK *,:,H,*,(B(B*E+9+JB#
K *E"9M()+*9"?,*1:NO#(B*,E"?,J)+9(2QP
<>R '7SUTWVXY ZC[\?,*,&()"*F]^(B$+*_()#%"J`M:_#(B*,E"?,J)+9(2QPOaVcb
d!+*,:eJ)'/fgZ
#()D_6,JBh&JB"#:i+*(2Q&J)!&j]^()#7&"*"?19a9"?,*1:iVcb]^,()&j_&+*k+()*1#/*1l+,9
#()*1E"?,J`9(B()#0 K *m,n6I+#%Y]a7Ic8"7?1#:F1^oh+?,&jPO(B*kE"9+Jk[\"9Dm?1J`p+*1:
(B#78+9%(`+*#^_&+De6,?rqcstStTuVXZvkV1@g,
]n'8"9w,(B#x[\+9Dm?,J)O()#^*,+w+JB]ncP!#
+6,6,JB()&c,J)0tyI"97()*1#%j*,&d,]n&c+*1*,+^~W]n| +9$"?
z;{ |{ } ~! v!V
,? #(B*,El,poh?,&jPi|1!|"|3 9%2D_#d! &c+?,#%Z,w#%()*,E"?1J`+9%(BQPFnl:1#a*,n#%D
F A+[51A[\+9D 0
<> ]n'8"9d]na&c+*l]n"9%~Wl|h+?1U,n(B*kE"9+Jk?,#(B*,E R +?,9%*t#%9%()#0h'6I*,:

a V V+ 1 V+ c
()*E"9()*,Em,()#h9%DP$~W| 9DGd, ]nAE"'
z { |{ } ~ vkV z { |'{ } ~ v!VZ z { |{ } ~vkV
V
z { |'{ } ~!vk4V+V z { |{ } ~1vkV+V c
p#61(B_J)!"k()*1E&"D_6,J)(B&c:;dD_"#%+[n,#i(B*E+9+JB#l9_9%,0 = &+J)J
Ie]^,*18"9SI+#*+*(Q:,9(B8(B8+"*&JB"#:&"*+?,9ef5d7,*

~
|
s StTuVXZvkV 0^ ~ | (2G,l'&61(B"* +[ d +J)J;,r~ | (B*kE"9+*,:,#ZIc8"O*k(2
:,9%(B8(28"#Tu+0 E,0 ,+#x+*N~W| *k(2 :,9%(B8(28"p [ X'dI#F1P+JBJ 8+*,(B#;0
7!?1#x]al,
8+
z { |{ } ~ vkV z { |{ } ~ v!V V
kPoh+?,&jP
#Z()*E"9+J [\"9Dm?1J`10
< "9AE"*,9+J)J2P"d(B[tStTWVXhI#Z R +?, 9%*x'6I+*1#()+*+[
StTWVX } TWV Vcb'X + *,:Nf()#wi#(BDe61J)&J)+#: +*(2&JB&j]^()#%r&"*"?,9Z]^,()&j &"*j+(B*,#CVcbA+*1:
*,F+,9^#()*1E"?,J`9(B()#n+[tSd1, * I~

I~
s StTWV"X^vkV s VVcb vkV &c+?,#%/+J)JM,YI+~ ,9;9%De#;(B*p, R +?19*5'6I*,#(B"*AIc8"/+*(Q:,9(B8(B8+#0
< &+?,#l, & _&(B*p(B#x,r+*,JBP$,()*1EiJ)')[ (Z[ 9w()*E"9(B"*;d,]a
&c+JBJ/(2l,
"!$#%+[xS mVcb
dY+*,::,*,$(2' & TS5@jVcb'X'0Ge&c+*!?1#
961,9+#%,+g 8+A9#%?,JBx#
Ss tTuVX7vkV ) *& ) ( TS5@jVcbX'
~W)| ,
y +97()*,#j+*,&+d,]aAIc8"+& ) ( T X +0
K *E"9M()+*9"?,*1:Dm?1JB(B6,J)Z#(B*,E"?,J)+9(2()#
< - ~ , ]#%, ?,6,6g, "#^IMYf()#A#(BDe6,JBh&JB"#:i*k(2&J)!&j]^()#%h&"*"?,9]^,()&j_*
&J)+##UDe"9%/I+*O"*1n#(B*,E"?,J)+9(2QPA+[ S@"JB. #t#cPr(2U*,&JB"#0# /i#(B*,E"?,J)+9(2(B#
V V c V 0Y7,*G]nIc8"
~

)
(
Ss tTWVX7vkV 1& TWS@jV X > 1& ) ( TWS@jV X
ccc 1& ) ( TS@jV X'
< 7,A6,9!+[()#7P:,&"D_6 "#%()*,Em,A&"*"?,9x+6,6,9%"6,9%(`JBP Tu#%A6,()&'?,9%
X0
γ
2
z
z1
z3
"*Q#%()D_6,J)w"97&JB&j]^()#%A&"*+?,9#
,
,
,
<- ] #%?,6,6g"#_ICf()#GD_"9%O&"~ D_6,J)(B&c: &J)"#%:&"*+?,9d]^,()&j*
&J)+##^"*1+97De"9%C#(B*,E"?,J)+9(2()#hV V cc V 0
-
γ
z1
z3
z2
<LK *G,+ 8+#%(B?I (B"*;d!]nIc8"
~
)(

s StTWV"X^vkV T w$& TS@V X
&
)( TS@jV X & ) ( T S@V XX

kP$:,&"De6g"#%()*,EO1&"*"?19x(B*F6,()&#0
< "9 E"*,9+J)J2P"d^]n*,: :,HI*1, *,+(B"* +[r0
!$
%# 0
K *1[\"9De+JBJBP"d,x]^()*,:,(B*,E*!?1Dm 9a[l&"*"?,9f+9+?,*,:$6 +()*aVcbh()#Y1
*!?1Dm 9n+[ ()D_#f]^()*1:,#h+*(2Q&J)!&j]^()#Z+9%"?,*,:VcbCT\1()#n*k?,Dr 9n&c*$ *,EM(B8+e(2[af E+#&J)!&j]^()#cX'0 D_"9F6,9%&(B#_:,H,*,(B()"*()#+#A[\"J)JB
]^#0
U+O+*P6I #j+9(B*,E$pVcbA+*,: ,+:,(B*,Ei"?p
]n+9: (B*1HI*,(2QP"0 Z#w"*1
[\"J)JB ]^#Z,(B#p6I;d j+F,O*k?,Dmg9A+[(BDe#pf,(2#p,()#p6, [\9+D 1
9()E+kdkDe(B*!?1#/,w*k?,Dmg9a+[;()D_#f,(B#/,()#/6Ii[\9"D ,wJB[ 0t7,()#/()#
,]^()*,:,(B*,E_*!?1Dm 9w+[f+9"?1*,: Vcb
@g(B^?19*,#Z"?1wIZ*,iDe%97]^IM
6IP+"?N&j,!"#%+d,1A]^(B*,:,(B*,EF*!?1Dm 9^()#^+J2]hcP!#7,A#De0
< 7,7]^()*,:,(B*,EC*k?,Dmg9+[gf+9%"?,*,:eVcb/()#Y#+De'()D_#U:,*,: 1/;vgT f@VcbX'0
7,ADe"#^E"*19J 9#%?,JBZ+g"?1^(B*E+9()"*"*&J)"#%:G&"*+?,9#x()#
<
CR ~ , , gr, _#(BDe6,J2P&+*,*,&': :,"De+()*;d *,:NJBpS
w[\?,*,&'(B"*i"* ]^,(B&j$()#n+*I+J2P!()&^"* '&61^nlHI*1(Bx*k?,Dmg9a+[
()#+J`:#(B*,E"?,J)+9(2(B#hV V cc V 0/71*]nIc8"

Ss UTWVX7vkV } ~ & ) ( TS5@jV X 1/;vgT f@V X
[\"9^+J)J &JB"#:&"*"?,9%#xfN()*N0
< 7,C6,9!+[+[ ,()#a,"9D ()#a 'P""*1:,C#&"6gp+[,(B#n&"?,9#%+dI9% k?,()9%()*,E
e&9%+()*N+De+?,*w+[U"6g"J)+E+P"dI,?pP9%P!(B*,Ei,De6,JB#x#?,&j #Z1l"*1
+ 8++*,A&c+*G A&"*8(B*,&:[t(B#h8+J)(B:,(BQP+0
< (BC,()#;,"9%DGd+*,Y&c+*&+De6,?Y,(B*kE"9+J"+[,SO"*A&J)+#:&+*k"?,9%#
'!9%D_JBP!?1()&jkJBP+dI6,9
8!():,:GI^"*1k*,
]^#Z+J)J ,A9#(B:,?,#^+[tS0
on"D_6,?1(B"*+[9#():1?,#
<LK *J)()E+kn+[,C9%#%():,?,p,+9DGd1(27()#a&JBc+9%JBPe+[5()*9#7mHI*1:r]hcP$
&"D_6,?1i9#(B:,?,#!k?,(B&jkJBP"0N7,$D_1: &+De6,?$9%#(B:,?,#m:,6g*1:,#
"*,A*I?,9A+[51#%()*,E+?,J`+9%(BQP+0
<>R ' #ZHI9%#%Z#?,616 "#%lIpSI~ +#Ze9%D_
8+,J)A#(B*,E"?,J)+9(2QPwVcb
0h7,*N1
R ?,9*kp'6I+*,#%()"* [aS+9|+!?,| *,: VcbCI+#w*,$*,E(28"m6g
]n9#C+[ZTuV+>Vcb'X'0
9DGd +*,:G#F,A9#(B:,?,Z_9%D_
8+,J)
K *6I9%(B&?,J)+9d!,9()#7*,
#()*1E"?,J`9(BQPe()#79%,0
<- ] #?,616 "#%CSI+#h# (BDe6,JBI7~ 6 +J)xnVcb
d#%~ l1 R +?,9%*h16,+*,#(B"*iJB+!#
J)()
StTuVX V Vcb bU TWV VcbX c
"
I~
7,9%#%():,?,G()# 0L IH *,:>(Bd/]aDr?,JB()6,J2P,G+ 8+16,+*,#(B"* P
V Vcb
I ~
~
StTWVXTWV Vcb'X b TWV VcbX TuV VcbX cc
j+k()*1EF,AJ)(BDe(2h+#^V Vcb
d]n"E I~
| |
JB()D StTuVXTuV VcbjX
7!?1#x]al&+*&"D_6,?1C,A9#(B:, ?,^#(BDe61J)p6g"J)#^kP
( TWS@jVcbX | J)(BD |W StTWVXTWV Vcb'X'
)
&
~ |
< y +97()*,#j+*,&+dI]al&+*&"D_6,?1 C,A9#%():,?1 l+[ ^ 1#%()D_6,J)C6g"JBA
kP
|
( T @jX |JB()D V
)
B
J
)
(
D
&
b
b
#%()*TuVX
#%()*TuVX
&"# TWV"X "
<- ]Ad#?,616 "#%xSiI+#p :,+?,,J)n 6 "JBhYVcb
d"I#~ p, R +?,9*'6I+*,#%()"*OJB+!#
J)()
StTWV"X TWV Vc b'X TuV Vcb'X Ibt~ cc
]a"* ^]n"9% dI &c+?,#%
- ] ,C6,9'8!()"?,#79()&j_F'!9+&'x,A9#():1?, StTWVXTWV OVcbX5:,!#U*,+t&"*8"9E"^+#tV Vcbc0 ]a8+9d"*1n&c*O#%(BJ)Jk9&
8"9
,A9#(B:,?,C]^(BO,(2^De +9 p]a"9 I0Y~ ADm?1JB(B6,JBP$StTuVXnPTWV Vcb'X StTuVXTuV Vcb'X TuV Vcb'X b TWV Vcb'X c
FE"'Z9%():+[ d,]n:,( 9*(` C +#():1#7]^(BG9%#%6 &x_V1v TStTuVXTuV Vcb'X X I~ bMTWV Vcb'X cc
vkV
I~ *,
] "*,A&+*G++AJB()D_(B#/F"1 +()* ( TStTuVX'@Vcb'X | J)(BD | v TStTuVXTuV Vcb'X X
)
&
vkV
.
< 1* 7&c+*_:,C,7#+D_hQP6g^+[g9(B&jl[\"9Y1()E",9Y"9:19Y6g"J)#@(B[ S$I+#/C6g"J)
+[U+9:,9xVcbx,*
( TSUTWVX@jVcb'X | JB()D | v TStTuVXTuV Vcb'X X
)
&
T ' X vkV ]n'8"9dI,(B#n[\"9%Dm?,J`l()# k?,(2C&?1Dm 9I#%~ "DepO?,#0 x#?IJ)JBP_(27()#hg%9
# $:1( 9*C]hcPN&"D_6,?1 T\+0 E,0kP &"D_6,?1()*,Ei,O*()9
R ?,9*kx#%9(B#X'0
< ,*NS I+#^+*#%#*k(`+J#%()*,E"?1J`+9%(BQP"dk19A(B#7*, !?1()&ji[\"9%Dm?,J)lF&+DF
6,?11N9#(B:,?,+0 [ * ,"*,JBP]hcPL()#F HI9#i&"D_6,?11N*()9
R ?,9*kx#%9(B#x+*,:1*G'!9+&^1_ ,TWV Vcb'Xh& _&(B*c0
<- ]Ad1]a$ J)J #%,
] 1 ]9%#(B:,?,C&+J)&?,J)?,#a&c+*J)#r p?1#:m&"D_6,?1' :,HI*1(B(B*E+9+JB#m+#m]aJBJ7+#m&"De61J)'&+*k"?,9O(B*kE"9+J)#0yI"9l()*,#j+*1&+d
]n JBJ A+1J)p_&+De6,?C#?1&jN()*E"9+J)#^+#
v b ^L&"#
T "X
v Y b
&"# T Xhv b Y v

v b ! UT"O X
'1+&JBPg@hJ)J/+[7,#()*E"9+J)#r+9$!9DeJBP:,( _&?,JBm #+JB8"P&"*
8"*()"*IJ5&j,*,( k?,#Z7k?,#p&"*"?19w()*E"9(B"*(B#w_6 ]a9[\?,J*1] !"J
_8+J)?,A9+J ()*E"9+J)#d +JB,"?,E"(2^()#^*,+ZFDi+E+()&p]n+*,: *,:N&+*,*,+
I+*,:,JBC8+9%PG#%()*,E"JBp()*E"9+J;()*1(B#%*,&+0
<LK #%,"?,JB:i]n+9*$P+"?()*+:8+*,&wIh,()#a()#nm9,9a:,( _&?1JBh#%&'()+*;d1,?
+J)#%$"*,m[Y,OD_"#%w()D_6 +9%j*kZ()*,m&"?,9%#+$0 #/"? #%,"?,JB: E"$19"?,E+
,A*,+#^&c9[\?1J)JBP$[ 9h,AJB&?,90
%
p'HI*,(Bp9%()E""*1"De'9%()&w(B*E+9+JB#
< 7,ZHI9%#%/QP!6 Z+[;(B*kE"9+J,]aZ]^()JBJ,&+*,#(B:,9a()#/A9(BE""*,"D_9(B&7()*E"9+JI+*
+*()*9%8+J +[J)*,E+ t@1IM^()#d *G(B*, E+9+J;[,C[\"9D
SUTu&+# #%()* "Xhv
,
]^,9%mStT\&"# #%()* "Xn(B#7#"D_A&"Dm1()*I()"*i+[U&"# T"X7+*,:#(B*T +X'0
< F()JBJ)?,#9w,ADe',!:;d1]aAj+l#Z*G1+De61J)
v b x &"#
T "X v!
< x&"D_6,?1Y,(B#5(B*E+9+J"?1#()*1EZ9%#(B:,?,&cJ)&?1J)?,# &j,*,( !?1#d,YHI9%#%#%6
()#m9%6,J)+&x,Z#()*1#7+*1:$&"#%()*,#n]^(Bi&"De61J)'e16g"*,*(`J)#n?1#()*1El1
[\"9Dm?1J`+FT\#A X
,

&"#cT "X
#(B*T +X

7,+ 8+l(B*kE"9+J ,*G &"D_#
~ v ~
b ^

<- ,] ]nmDe+l1m&"D_6,JB'G#?1,#%(B?()+* V d #vkV v 0 (B*,&
G
*,E"#l[\9+D G Yd1e8+9()+,JBFVN&JBc+9%JBP 9c8"9##r1i&?19%8"iV
9+
) YdU]^,(B&j ()#A,$?1*,(Br&()9%&JB
V i9c8"9%#:L+*,&*k(2
&J)!&j]^()#%+07,A()*E"9J ,*g&"D_#
z { |{ } ~ vk~ V , T" ~ X ^
]^,()&j]n&c*]^9%(B(B*$9D_#x[tV +#
z { |{ } ~ v!V ~ ,"T VX ~ |~
^ Vx
7,()#^#(BDe61J)(BH,#a
z { |{ } ~
,
+vkV
VZ V+t "9
z { |{ } ~ vkV
V+U VZ
< #?1DeDe+9(B#%[u+9dY]nGIc8"G&"*8+9%: :,'HI*,(2()*E"9Jh+[w9%c+J
9(BE""*,+De'9(B&'6,9##%()"*O(B*kCw&"*"?,9U()*E"9+J!+[ p9(B"*I+J6,9##(B"*
"* &J)"#%: &+*k"?,90 - ] ]a?,# 9#(B:,?, &cJ)&?1J)?,#$L&"D_6,?1 ,()#
()*E"9+JW0
< 7,CHI9#Z#6N()#hO[u+&"9(Bp 1:1*,"D_()*,"9
V Vx TWVx ! XTuVZ>^ ! "X
#_"?,97()*E"9+J;g&+De#
z { |{ } ~ SUTWVX7vkV

]^,9%
StTuVX TWVZ ! X TuVw>x ! +X
< he,()#F6g"()*F]nG&+?,J):>?,#G6I+9(`Jh[\9+&' ()+*,#O(B[w:,#%()9:;dh,?e]a$
J)J^?,#%
9#%():,?1_&+J)&?,J)?1#C()*,#c:;0F7,O[\?,*,&()"*StTuVXpI+#A#(B*,E"?,J)+9(2()#p wC
! +*,: p ! 1d,?1A,F&"*+?,9 V e"*1JBP E+#l9"?,*1:,OHI9#
#()*1E"?,J`9(BQP+0 (B*,& +?,9&"*"?,9()##%()D_6,J)N+*,:+*(2&JB&j]^()#%+d^]n &+*
&"D_6,?1p,A()*E"9+J;?,#%()*,EF,A9%#(B:,?,C1"9%D +#
) *& ) ( TS5@ p^ ! X
< p&"D_! 6,?1a,h9%#%():,?,a]nh,
8+hp#%h]^IMYk()*,:m[ #()*1E"?,J`9(BQPlS_I+#Y! p^
10Y7,:,*1"De(B*I"9h+[tS &J)+9J2P$I#^_#%()D_6,J)p9%iw^ Tu"*,G&c* "*,J2P :,(B8!(B:,"?1e"*,G6 ]a9i[AVO ! g[\+9G &"D_()*,E
:,()#%&"*(B*k?,"?,#X'dx#LS (B#JB[AI+# #%()D_6,J)G6g"J)0 7k?,#$]a &c+*?1# 1
[\"9Dm?1J`
)( TS@Vcb'X | J)(BD |W StTWV"XTWV VcbX
&
#FI^+?,97()*E"9J;()#7 !?,+J | J) (BD StTWVXTWVZ ! X

|
J)(BD
VZ ^ !
)
! 7!?1#x]al,
8+A&"D_6,?1:

)
v b ^L&+# ! <- +rIMA8"*,"?,E+ &"D_6,J) *k?,Dr 9#CDe+:,O+* 6,6 +9*,&O:,?19()*1E
,CD_():,:1J)w[+?,97&"D_6,?1(B"*;d!,'P:,()#+6,6gc+9%:Gh,C*1:;07,()#7I#
GI+6,6g*;d+[7&"?,9#%+d#()*1&i(2[n+*,e(B*E+9#l9+JY[\?,*,&'()+*"*9%c+J
()*9%8+Jh"*1&c+*L"*,J2PE"'e 9cJ7*!?1Dm 9 K [Z"?,9O&"De61?1j()"*+[Z,()#
()*E"9+Jg*1:,:?,6$]^(2+* #"D_]^19p,*$]aCk*,
]Ia]n 8+De+:,
FD_()#j++0
< 7,()#h&j,*,( k?,A"*,JBPe]n+9k#7]^,*P""?G()*E"9p+**G(B*98+J;+[5J)*,E+
Yd +9w+[#%"DerDm?,JB()6,JBA+[a Y0AyI"9Z()*,#j+*,&+d (Bp&+*,*,C:,()9%&'JBPGI*,:,J)
,A()*E"9+J
v
b ^ &"# &c+?,#%]^,* "*,De++#1&jI*,E"[r8+9()+,JB#GV "*1"*1JBP
*,:,#O?16]^(B #D_(2&(B9&JBe(B*,#%c+: [w&()9%& JB_A*,O &JB"#: &"*"~W?, 9
T *,&c+*G9%P,&jI+*,E"+[t8+9%(`+1J)#^V ()*1#%+:;d ,?1x,xI+#x(B#
]^*_6,9%",J)De#U &c+?,#%^"*,^#%+9%#Y*,&"?,*9()*1El1619#%#(B"*,#/J)() 4V d
]^,()&jGIc8"*,"*()#%"J`:#%()*,E"?1J`+9%(B(B#X'0
9%P
n
]
'
"
8
9

d
#
"
_
D
)
(
_
D
Z
#
"
,
*
m
&
+
*
I
+
,
*
1
:
)
J
r
,
%
#
O
'
,
6
9
#
%
#
)
(
"
,
*
A
#
P
$

#
!
P
e
D
_
D
<>
+9E"?1De*k0yI"9n(B*,#%+*,&d11p[\?,*,&()"*&"# r()#n#%P!DeD_9(B&C+9+?,*,: Yd
#l,p 8"Z()*E"9JI()#/ k?I+J I+J2[;+[; ,x()*E"9+JI[\9%"D AO t0t7k?,#
v !
b ^ &"#
c
< no I+J)JB*,E++-^#m(B[YP+"? &c+* &"D_6,?1,#O(B*kE"9+J)#x]^(B,"?1w?,#(B*,E$&+DF
6,J)D_1:,#lTu+*,:]^(B1"?1^?,#%()*,EeN+61J)
X0
K *E"9M()*1E_9(B"*I+Jg[\?,*,&'()+*,#^"*,A9+J J)(B*,
< - ]]aA&"*1#():19x k?,(B:1( 9*7QP6gA+[5()*E"9Jd1*,+DeJBP_,C(B*E+9+J
+[tO9(B"*I+Jg[\?,*,&()"*eT X i"T Xh"*,A*()9A9%c+J JB()*,
eT X v

i"T X
w*G'1+D_6,J)C(B#

v v
t
T - IMO,(B#_(B*E+9+Jh&"*8"9E"#-[\+9 G"9 G"*,&c+*L?,#%
,O&"De6,+9(B#"* #%A?,#%()*,E d+*,: [\"9 F"*,F&+* ?,#O1
&"D_6I+9%()#+*$#x?,#(B*,E
X'0
< OI*,:,J)Z,C(B*,:,H,*,(Bp()*E"9Jdk]np]^9%(Bw(B7#7mJ)(BDe(2a[:1HI*,(2p(B*'
E"9+JB#J )(BD v
A&c*G]^9%(Bp,()#^#
J B()D s StTWVX7vkV
|~~
]^,9%f (B#^,A#%9+(BE"7J)(B*,p[\9"D & & *,: SUTWVX 0

< 7,p[\?,*,&'()+*GSN()#n#%"D_]^I^J)+9E+w*,+9h,w"9(BE"()* T\[\"9h(B*,#%+*,&+d StTuX
X1?1n:1&9%c+#%# k?,(2 !?1()&jkJBPe+#a"*,xDe
8"#7c]hcPF[\9+D ,x"9(BE"()*;0hTWyI"9
()*,#j+*,&+dSUT
Xd StT lcX'dSUT
$ X'd +*,:StT $ Xw+9%O+J)J59+?,E",J2P k?I+JU
14+"1"04X'0
< O*,
] ?,#e$8"9P 8"9#(BJ)m&j,*,( !?1ek*,
]^*# % # @
,p(B:,cm(B#am9%P_O61?,#1p()*E"9J c]n
Pi[\9"D 1C"9(BE"()*;d]^,9C(2h()#
J`+9%E"+dk"?1/
]n+9:e()*1H,*,(BQP+d!]^,9x,Z()*E"9+JI(B#a#%DiJ)J0 K *i+?,9a6,+9%()&?1J`+9
&c+#%F]a_#I+JBJt#1(B[ A,F&"*"?,9A"*NJ`9E"O#%D_(2Q&()9%&J)@;,(B#C()#Ck*,
]^*
+#7,AD_1:+['6I+*1:,()*,EF#%D_(2Q&()9%&?,J)+9n&"*"?,9%#0
+

[\9"D
<>R ' :,*,A,#De(Q&(B9&J)AV & - ,(B#7&"*+?,9^I+#7,A#+DeC*,:,6g"()*#Z#^f 0
&
m& @
+
CR
-R
γ
R
R
<LK [ZSLI+: *1 #(B*,E"?,J)+9(2()#d 1*]a$&"?1J):?1#oh+?1&jkPA"?19#MON#%,(B[ ,A&+*k"?,9x+*,:&"*,&JB?,:,AIM
U
S
W
T
V
7
X
k
v
V
s
SUTWVX7vkV1
]n'8"9d,9%w9x#"D_^#()*1E"?,J`9(B()#d+*1:e#%l19Z+9%x#%"D_x&"99%&'()+*
9%De#7&"D_()*,Em[\9%"D9#(B:,?,#0
<>R ' #7HI9#x]a"9%"?17]^,9A,A#(B*,E"?,J)+ 9(2(B#n[YStTuVX'07,'P&&?,97]^,*
V
V l
V B
V , , ,
V
7,()#l+9_,F[\"?,9&9%"##%#m+*1$+g
8"i:1(`+E"9+D T ]n$&+*##?1De_I
& ()#x#$J)+9E+Ix,l#D_(2&(B9&JBC&"*+()*,#x,l?,6,6g9xQ]n$#%()*,E+?,J`+9%(B()#d
#()*1&C]n 9%l'8"*k?I+JBJBPE"+()*,Em_JB & X0
<- ] ]al?1#A9#():1?,A&c+JB&?,JB?,#h_&"D_6,?1p,A(B*kE"9+J
StTWVXvkV1
s

7,()#x&"*"?,9^]^()*,:,#w+9+?,*,:,#()*1E"?,J`9(B()# +*,: "*1&m*k(2
&J)!&j]^()#%+dI,?1x:,!#^*,7]^()*,:+9+?,*,:,A+,97Q]ae#%()*,E+?,J`+9%(B()#nZ+J)JW0
7!?1#

StTuVXhvkV *& ) ( TWS@ X 1& ) ( TS@ X
s

< e&"D_6,?1l1 #l 9#():1?, #d ]a m*,:NeHI9#ZHI*1: "?1Z,l*I?19+[t1
#()*1E"?,J`9(B()#h *,: 0

< 7,[\?1*,&()"*V l(B#79$M *,: dI,?1Z(B#7:,9%(B8(B8" V ()#
*,"*9%rM +e[ ,#%Z6,J)+&#0t7k?,#nV x,+#/A#%()D_6,J)79%rY,#%
6 "(B*#d#% StTuVX ,TuV XpI+#lG#%()D_6,J)O6g"J)0O_&c+*k?,#?,#F1

#()D_6,JBp6 +J)w[\"9%Dm?,J)r[\"979#():1?, #-

( TS5@ X | J)(BD V
)
&
V
+, *!?1De9"9+*,: :,*,"D_()*I"9i*,: 9,dZ# ]aDecP?,#%

R ,"61(Bj1J #h9?,JB+ ( TWS@ X | J)(BD )
&
V+
c

#()D_()J)+9n&"D_6,?1jM()+*E"(B8+#
( TWS@ X
)
&

!
! | )J (BD V

!
! < on"Dm,(B*,()*,EFJ)J ,#%l&+J)&?,J`M()+*,#d! ]nAE"'

StTuVX7v!V jT s ! !
< A&c*G!?1#x&"D_6I+9p,Cf ()* E"9+J;+*,: ()*E"9+J +#
t
S
W
T
V
7
X
k
v
V

StuT VX7vkV1
s
! < U+k(B*,EFJ)()D_(B#dk]nAE" '

v
vkV

B
J
)
(
D
! V < m#%?,D_Di+9%()Z#%m[u+9dk]n 8+ADi+*,+E":_m#%,(B[ a,w&"*+?,9n[\9+D ,p9%c+J
J)()*1hmA,(BEl#%D_()&(B9&J) dk]^1()J)^6,(B&jk()*,E?,6$#%"De^9%#(B:,?,#aj+JB()*,E ~ 0
7,m*,p#%6 ()#^$#%,
] ,l()*E"9+J5"*N,()#x#De(Q&()9&J)+&'?I+JBJB|PE"~ !#
i9%+# & 0Z7IM #p &c+?1#OV$(B#x8+9%PNJ`+9%E""* d;*,: ()#
8"9P#De+J)JW0

V & d,#%
< "9A6,9%&()#JBP+d,]^,*Ve(B#7"* d V &md +*1:G#%
& V &
#

&t V t &
7,A&"*+?,9 I+#^J)*,E+ &md,#F]nIc8"
vkV &
V Y &
7,9(BE"%Q,+*,: #(B:,$E"!#m 9+# & d/+*,: #% kP,#k?,
,"9D ,AJB[ %I+*,:#%():,CDm ?1#%Z+J)#%FE" OF9,07k?,#^]nIc8"
v
!
wE+(B*;d1*1+C,^,CHI*IJ +*,#]n9Z(B#^_9%c+J;*k?,Dr 9d +#^(B^+?,E"7_ 0
< P#%P!D_De'9P"d,]a&+*+J)#%_(B* E+9p ,(B#h[\?,*,&()"*"*GI+J2[`QJB()*,#
v b t !
< 7 ,()#xI+#%()&#%9E+P]a"9k#Z]^1*,'8"9C+*,l(B#^()*E"9()*,EeF9()"*IJ;[\?,*,&'
()+* iT" XeT Xd/]^,9 +*1: +9%6 +JBP!*,"D_(`+JB#dU[\9%"D 6,9
8!():,: I I+#A$#+De']^IAJ`9E"9p:,E+9OI+* _0 PN96gcM()*1E
,+g
8"+9E+?,De*^"*,A&c+*G#1 ]
|

< hK [ | +*,: +9%C6 +JBP!*,"D_(`+JB##?,&jGI7v T rX v T lX Ld
,* vkV e# & 0

< 7,()#^J)DeDer(B#^+J)#%F8+JB():i[\+9h,AJ)
||%]a 9^#De(Q&()9&J) 0

<LK [pv T lX v T lXY d,* ()#O*, +_ (B*k ~E"9+,JB$"* ,9%c+J7JB()*,
Tu(Br:,&cP#mN#J)
]^J2P,X0 yI"9r(B*,#%+*,&d :,!#m*1+O&+*k8+9%E"TW
()*1HI*1(BQP"d1(B #7J)(B+C(B*kE"9(B*,Em1A[\?1*,&()"* X0
K *E"9M()*1EF9(BE""*,"D_9(B&'Q9(B"*I+J,[\?1*,&()"*,#
< - ] JB #h9P$1#+D_A&j,*1( k ?,AF ()*E"9M

v
"
< A&c*G]^9%(Bp,()#dI+#^g[\"9%+d +# |
J )(BD s V+t vkV
|
< A&c*&"De61?1
vkV
s V+Y ?,#(B*,Ew9%#(B:,?,/&c+JB&?,JB?,#@+()*A,(B#&+#d"(B?19*,#"?15w k?I+J 0 xx[\"J)JB ]
,F6,9%8!()"?1#l6IM9*;d]ae*,
] ?1#%Ic8"_G#,
] IA, (B*E+9+J
E"#h_9,0
< Z*1[\"9?,*,JBP"d,9A()#7*,
] O#%*I+E, dI g | &+?,#
vkV
|
V
&+*E"
:, | #i *,$E"9 # &
07, 6,9%",JBD ()#FIM
' ! 9%D_ JBPGJ)+9E""* 0wy +9Z(B*,#%+*,&d;(B[tV *&mdg]^,()&j (B#Z"* dg,*
d1]^,(B&jNg&"De#78"9PJ`+9%E"C+# & 0

< E"r+9+?,*,:,()#6,9%",J)D ]ne*,:?,#_, ' [\9"D & & 919AI+*,_?,6,6g9#De(Q&()9&J)0e5NDe++F#%?,9_I
,(B#h]^()J)Jg]n+9 d J)^?1#^H,9#%x&j ,| &jI
v!V
e+# &

V+U
V & dI#%
<LK [tV_()#7"* d,,* V &mdI#%

| & V & "
I+g"?1 | ?1# oh+9% #%(`+ *&Q"9%:,()*, #$V gdw+*1:
&"D_6,?1

]n'8"9d#(B*,&m]aF+9r"* , ^#%D_(2Q&()9%&J)d 1d;+*,: # "0
7!?1#m,*k?,D_9"9rI+#mDe+E"*,(2?,:1$OD_"#$"d/+*,:,$:,*,"D_()*IM"9
I+#^DiE"*,(B?,:,CM^J)c#% & "d,| #

Vt &p
c.

()*,& I+#^JB*,E+ &md,]nC!?1| #ZIc8+
vkV
&

V &
7!?1#x,A()*E"9+J "*,AJ)'[ %I+*,:#(B:,AE"!#hF9e+#
&
0
R
-R
γ
R
C
R
< A*,
] &"D_6,?1C,A&J)+#:&"*"?,9^ (B*k | E"9+J
z
vkV
s V
| ~

R 'CSUTWVX 077,[\?,*,&()"* SI#Z#%()*,E"?1J`+9%(B(B#^wV %d ,?w"*1JBP
,Z#()*1E"?,J`9(BQPO t()#/9J)'8+*k0 ()*1&Zf ]^(B*,:,#h+*,& 0 %
|
+9"?1*,: %d ]aAIc8"
z vkV p*& )( TS@. X'
s V+U
|
()#*,"*9mM d
< 7,x[\?1*,&()"*eV xI+#a#%()D_6,J)79r %| d1+*1:
#iStTWVXn,+#x_#%()D_6,J)p6g"JB+d1#%
& )( TWS@ QX | J)(BD TWVZ X
| I~ V+ U

|
JB()D V | p

7!?1#x]al,
8+

z
vkV s V
<- ] ]a_++_J)(BDe(2#Z+# & 0O $ 8"e+J)9%c+:1P ",#9%8+:IA1 ()*E"9+J E"#h_9,dI#%F]a$ 9J)'[ 7]^(B
v

<LK *E"*,9+Jd]^,*()*E"9()*,Em +* 6,9##(B"*N#%, ?,&j #

iT" X v iT" X
]^,9% +*,: +9%76 +JBP!*,"D_(`+JB#U*,: (B#YC*,"*9%A&"*,#j+*dk,*F"*1
#,"?1J):G?,#A,A?,616 9Z#%D_(2Q&()9%&?,J)+9n&"*"?,9^]^,
* _+*,:1lJB ]a9
#D_(2&(B9&?1J`+9^&+*k"?,9x]^,
* 10FT +#%()&cJ)JBP+dI,m(B:,c_()#x$6,?1# 1
&"*+?,9al]^,98+9h,w(B*kE"9+*,:e()#/#De+J)J`X'057,9%p()#alE"*,9+JgJ)D_De1
< R | +*,: O6g"JBP!*,+De()+J)#^#?1&jIv T lX v T rX "d +*1:JB O*,"*9_*!?1Dm 90 K [ 1d,*
iTuVX
vkV e# & @
e| TWVX
+*,:(B[ 1d,,*
iTuVX
iTuVX v!V _# & @
c
< 7 ,O6,9!+[+[,()#x[u+&C(B#C$,(2w:,()+?,#d *,: ]^()JBJgm"D_(B%: 0_T *,mI#
:,(B8!(B:,$,&"*"?,9F(B*k Q]n6,()&#dh"*1$]^,* TuVXO()#O#De+JBJdY#cP
TWVX ! &md *,:,l,9^]^1
* TWV"X^(B#xJ)+9E+
X'0 - CIM "*1JBP
*,:,#alIc8+pA:,E"9%x"*,x,()E+,9Y,+* Fdk +#/| "6,6g"#:eAQ]ar,(BE",9Y[\"9
,A6,9%8!()"?1#ZJBD_Di10U7I #x &c+?,#%,
[u+&"97,J)6,#^FDe++C1
()*E"9+*1:#De+J)JB90hT K [ ,+#^m:,E+9C k?I+J O"9hJ)##nI+*Fd!,*1
()*E"9+J ()#7*,+^&"*8"9E"*kjX'0
< *1&+*+J)#%OI+*,:,JBC9(BE""*,+De' 9(B&'9()"*IJI'6,9##%()"*,#x#?1&j +#

&"# T X v U
()*,(B#7Di*,*,90 *,C]hcPF:,F 1()#7()#hF?, #A,C[\"9%Dm?,J)
&"#cT X
F#6,JB(Bh,()#^(B*kE"9+J;(B*k mQ]ne6,()&#d
v m v t t +*,:]n+9$"*+&jG6,()&A#6I+9MJ2P T ,pHI9%#%^"*1?1#()*1Em,A?,6,6g97#%D_(2
&()9%&JB+d,w#&"*,: "*1w?,#%()*,E,ZJ)
]n9n #%D_(2Q&()9%&J)cX'0 ?1a,(B#aj+# !?1(B
]^,(BJ)0 k?,()&j9 ]hcP"d()* 1()#A&c+#% +~ d()# ?,#%G",#%98"eI~ &+# T Xp()#
,A9+J;6I9%x+[
Tu"9 Xd,# ()#h1 9%c+J 6I+9%x+[ 07k?,#

&
+
#
T X v & v

()D_()J)+9J2Pm]nIc8"

#%()*"T X v
t U v 1@
~
+JB,"?,E" ,()#C(B#CDe+9Oc#()J2P 6,9%
8"*P ",#%9%8!(B*,EGI (B#A+* :1:
[\?,*,&()"* 0 P#%P!D_De'9P$]n +JB#FIc8"
&"#
"T X v
b t <> ]n'8"9d/"*,i,+#rN ,(Br &c+9%[\?,J/]^,* :,+J)(B*,E]^(2&"D_6,J)k
6,9##(B"*,#^#?1&j +#

&"#cT X v

F Y8+* 1"?,E" &"# "T Xx()#x,r9cJU6I+9p+[ d ,m()*E"9+*,: (B# ,A9+J;6I9%x+[ dIg&+?,#% m Y(2#JB[I+#79%c+J+*1:(BDi+ E+()*I+ 9PF~ 6I+9#0
Tu7,p9%c+Jg6I+9h+[ ()#n#% "D_~ 1()*,El~ Dm ?,&jD_##%()9d1*,+DeJBP X'0
K * ,()#w&c+#%m,Og#C,(B*,E$ :,(B #p?, #m,F+J29%*IO+6,6,9%+&j+g
8"+d
*I+D_JBPe#61J)(B%(B*,Em&+#
T X +*,:]a"9k(B*,EF"* +,C?,616 9
+*,:J)
]a9^#De(Q&(B9&J)#0
Y9(B*,&(B6I+J 8+JB?,C()*E"9+J)#
<- ] ]al?19*e(B*E+9+JB#^]^1()&j 9*,+cdI#9%()&JBP#%6 +k()*,E1dg&"*8"9E"*kd
,?1m]^,(B&j &c+* #()JBJag$'8+J)?I: (B*]^I.
#O&+J)JB: %619()*1&()6,+J/8+JB?,
#*1#+0/7,#()*E"9+JB#h&9%"6?,68+9%P*1 ] +*,:1*;d,[\"97(B*,#%+*,&C()*$1
"?1*,:I+9P Ic8!()"?,97
[ 70
< w#x+*'1+D_6,J)dI&"*1#():19h19%c+J (B*E+9+J
I ~ v
<- +(28"J2P"d,+*,C]n"?1J):'6g&Ix~ 1()#7()*E"9+J _ k?IJ
JB* ~ J)*;T"X JB* T
X J)*;T"X
,
<> ]n'8"9dY#%9(B&J2P #6gc!, (B*,E,d ()#A*,r(B*kE"9+,JB_+*1e(B*k98+J &c+?,#%l[,l#%()*,E+?,J`+9%(BQPix10wTu7,C[\?,*1:I+D_*j+J;,+9D [U&+J)&?,J)?,#
:,#F*,+_+6,61JBP+
* h[\+9m,#De9%c+#%"* X'0 719G()#m+*>(B*1HI*,(2
+De+?,*l[7+9%c(B*,FHI9#O+*,:[\"?,9 !?,+:,9*k#dY*,:,e6,9##(B"*
(B#n()*1:,HI*1(B0 &+?,#%[ ,C#%()*,E"?1J`+9%(BQP"dk1()#nQP!6gC+[5()*E"9+Jg()#
#"D_()D_#h&+J)JB:G $# $ $ 0
~
< p#61(BC#%()*,E+?,J`+97(B*E+9+JB#h ()*,EF*,+*Q&"*8"9%E"*dg,9%(B#7#%()J)J;F&9j+(B*
#*1#O()*]^,()&j ,r(B*kE"9+J5+[ [\9"D m_ k?I+JB#pJ)*;T"X0wr:,HI*1
, # Z+[ai#%()*,E+?,J`+9Z()*E"9J5 k?I+JU,OJ)(BDe(2cd +# 1d

+[,p(B*E+9+Jg]^,9%l*()*9%8+J [9+:,(B?,# r+9"?1*,:$,C#%()*,E+?,J`+9%(BQPFI#
*G9D_
8":;0 K *,()#^&c#+d,]a Ic8"
I ~ v JB()D I~ v v

b
< ()*,&n]ahIc8"7&?1Y"?1U,h#%()*,E+?,J`+9%(BIQP+~ dM,h9(BE"%I+*,:m#%():,a()#t&"D_6,?1j,J)
+#
JB()D b JB* LJB*

J)(BD b JB* JB*Cn J)*x JB* JB*x
I~ 7!?1#x,A6,9(B*,&(B6I+J 8+JB?,C+[ J)*x0
< 7,()#76,9%()*,&()6IJ 8+J)?1C:,HI*1(B(B"*()#nDe"9%C#?,J)pI+*, (BhDicPiH,9#%Z+616 +90
K ^()#^()D_6 +9%j*ka, IMx,A()*9%8+J;+*,A9D_
8"#w+9%"?,*,:,A#()*1E"?,J`9(BQPe()#
#%P!DeD_9(B&+0 ?16,6 +#m]aO9D_
8": ,m(B*k98+JhT +X7[\9"D ,r"9(BE"()*
()*,#c:G[7T +X'd,+*,:,*JB 10tA]a"?,JB:G,*GE"'
I~
JB()D b J)* LJB*

J)(BD b JB* JB*CaLJ)*x JB*x 1
<LK [ ,9%^+97Dm?1JB(B6,J)/#(B*,E"?,J)+9(2(B#dM]n^9%D_ 8+^+*F6,#%()JB"*O()*9%8+JI9"?,*1:
+J)J +[,D #%()Dm?1JBj*,"?1#JBP+d,+*1:G,*GJ)'^6,#%()J)+*E"O_9%,0
on"D_6,?1()*,EF6,9%()*,&()6IJ 8+J)?1C()*E"9+JB#7?,#(B*,EF9#():1?,#
< _*,
] #%,
] ,
] G&"D_6,?1F6,9(B*,&(B6I+JU8+JB?,F()*E"9+JB#C?,#(B*,EG9#%():,?1
&c+JB&?,JB?,#dg?,#(B*,Ei]^IM #wk*,
]^*+#Z,rD_,:N+[Y()*1:,*: &"*"?,9#0 Z#
+*'1+D_6,J)d1]n:,De"*1#%9C,A&"D_6,?1(B"*+[
v

- +wIMn1()#n(B#7r#%()*,E"?1J`+9/()*E"9J g&+?,#p,C()*E"9+*,:$I#7m#(B*,E"?
J`+9%(BQPi 0
!
,
,
<LK [h]ne:,():1*
lIc8"e,e#()*1E"?,J`9(BQP+d ]n : ?,#r9%6,J)+&_,e:,"De+()* +[h()*
E+9()"* [\9"D T XpT & &AXl*,:1* J)''& +#rg[\"9%+0
e&c+* #()JBJY:,NIcdU,?1O*,
] ]a$Dm?,#F+JB#N9%D_
8"+* G+9"?1*,:1
#()*1E"?,J`9(BQP +*,:JB 10 K *+,9l]a"9:1#dU]ai&+*9']^9(2F,i+ 8+
()*E"9+J;+#
v v

J)(BD JB()D b U
A&c*9%]^9(2p,()#^+#
J B()D J)()D b s SUTWVXhvkVZ s StTWV"X^vkV
| | ]^,9%AStTWV"X d!f ()#/,p#9+()E"nJ)()*1x[\9%"D & !d,*,:if ()#h1#9+()E"^JB()*,w[\9%"D r &m0
+
+
CR
+
Cε
-R
γ
−ε
ε
-R, −ε
γ
R
ε, R
< p?1#a,n9%#%():,?,a,"9D ]ah*,:mZHI*,:mw&JB"#:r&+*k"?,9t#%"D_]^,9+0
"*,r&c+* 6I+9()+J)J2P&J)+#r,O&"*"?,9d
P +:1:,()*,Ee,O,(BE#De(Q&(B9&J)
,?17FDe++OE"*!?1()*,JBP$&J)"#%:G&"* "?,97"*,+JB#O*,:,#^ 1| #%DiJ)J #%D_(2
&()9%&JB 0xT *,C&"?,J):J)#r?,#p,CJ)
]a9h#%D_(2Q&()9%&J) (B *,#%c+:[ d
,?1p+*,mI+#Z$?,#% +*,: *, &c+?,#%m+,9]^()# ]^()J)Jg&"De
[u+9h!FJ`+9%E"+0
< A&c*&"De61?1p,A&J)+#:G&+*k"?,9^()*E"9+J
z

s s StTWV"X^vkV
?,#(B*,EF9#%():,?1&+J)&?,J)?1#0y + &"9 (B*,E_S +# |

StTWV"X VITWVZ XTWV QX
]n#I$S,+#i#%()D_6,J)6g"J)#iM$1d %d^+*,: %0 [p,#%+d7"*1JBP 1
#()*1E"?,J`9(BQPw ()# (B*,#(B:,U,&"*+?,90 ()*,&Y,Y&"*"?,9(B#+*(2&JB&j]^()#%+d
]nA#I7,+ 8+l(B*kE"9+J;(B#7 k?I+J; |
Tu
|

)
(
*& TS@ QX I~ J)()D VITuVw QV XTWV+Q X QX

"T X
)
7!?1#

t
S
u
T
V
7
X
!
v
Z
V

t
S
u
T
V
h
X
k
v
w
V

U
S
W
T
V
7
X
k
v
V
t
S
u
T
V
7
X
k
v
V

s s
*,: 10A7,OHI9#+*,: ,()9%: 9%De#
- ] ]amj+OJ)(BDe(2#Z+# &
+9m]^IMC]nO]n+*c0O7,O[\"?,9 9D E"#p9P+"9%:I+* #AJ)D_De10
7!?1#x]al,
8+

v
StTWV"X^vkV
g J)(BD b
< OHI*,(B#G"?,9^&+De6,?j()"*$]aA*,:OHIE+?,9A"?17]^, StTWV"X^vkV10
<LK [h]ne]n9$(B*kE"9(B*,EG 8+9r,i*()9%i&J)!&jk]^(B#e&()9%&JB +[79+:,(B?,# !d
()*,#c) :l( +[ ?,#t7&JB&j]^(B#/#D_(2&(B9&JB d ,*l1()#()*E"9J+]n+?,J): k?I+J
*& TWS@jXCP,$9#(B:,?,i,"9%DG0 (B*,&_]ni"*,JBPIc8+I+JB[h+[h1
&"*+?,9d"*,wD_()E+ka) E"( ?,##nIa,p(B*E+9+J "*i,p#%D_(2Q&()9%&J)^#%,"?,JB:$ +6,6,9%
(BDiJ2P 1& TS@j"X'0Y7,(B#x(B#7()*,:1:1&+#~

< ~ R 'UStTWVX gn[\?,*,&'(B"*C]^(BA7#%()D_6,J)6 "JBYVcb
dM+*1:JB A+*1E"J)#0YyI"9h+&j !d,J)'hf :,*,+A,A+*(Q&J)!&jk]^(B#wVIT %X Vcb5 d
07,*

~ ) (
J)()D b s StTWV"X^vkV T X *& TS@Vcb'X'

< !()*,&7StTuVX,+#tx#()D_6,JB/6g"JBnUVcbcd+]an&+*r]^9%(BnStTuVX TWVX ,TWV
Vcb'X'd ]^,9 ()#pi[\?,*,&()"*]^,()&j ()#w*,"*9%+*,: +*,+JBPk(B&mpVcb
0y 9%"D
UcP!J)"9
#7[\"9%Dm?,J)

TWV"X TWVcbX TWVcb'XTWV Vcb'X c
]nAIc8"A1 R +?19*x' 6I*,#(B"*
StTuVX V TWVVcX b V TWVcbVcX b TuVcbX c
+*,:# & ) ( TS@jVcb'X TuVcbX'0 !_]aAIc 8"AF#,
] I
~
W
T
V
X
J)(BD b s V Vcb vkV T X TWVcbX

l:,,()#d!]awDe+x,Z&j,+*,E"Z+[ 8+9%(`+,JB#/V Vcb dvkV v d
O?,9%*G,AJ)'[ %Q,+*,:#(B:,A()*
X TuVcbt J)()D b v ]^,()&jG#(BDe6,JB(BHI#n
Xhv
kJ)()D
u
T
c
V
t
b

b X
()*,& ()#w+*I+J2P!()&lAVcb
d (Bp(B#w&"*()*k?,+?,#d+*1: # TWVcbt TWVcbX
+# 10p+k()*1E$J)(BDe(2#lTu&'[30w,r6,9!+[+[aoh+?1&jkP A"?19#MA()* " X
,AJ)'[ %I+*,:#%():,Ag&+De#
TuVcbXhv ]^,()&jG k?I+JB#7,A9(BE"%I+*,:#%():,0
<LK *L+?,9m&c#+d]aG,
8+G &J)!&j]^()#%G#%D_(2&()9%&JB_[\9%"D +*1E"J)# 1dY,?
]np&+* ?,# I()6e,p#%()E"*$+*,:&+*,#(B:,9n,p9%8"9#%:+*(2&JB&j]^()#%w&()9&J)
d,]^,(B&jE"!#7[\9%"D+*1E"J)#^O t0 P$1JBD_Did

J)()D b SUTWVX7vkV *& ) ( TWS@jX
"
|

|
1J) (BD V
b V >
V

?(B*,EO,(B#7I+&j(B*kO"?,9^6,9%8!()"?1#x&+J)&?,J`M()+*,#d!]nAE"'

U
k?,#

v t T X %
< w#x+JB]ncP!#dI"*,A&+*G++A9%c+J+ *1:(BDi+E+()*I+9P_ 6I+9%#h_&"*1&J)?1:,

&+#
T X7v
U

(

1/tT X7
v /T3 X'@
U
#()*1&C,AJ)9h(B*kE"9+J;(B#7 8"*;d ]nAIc8" (
1/tT X^
v T X
b .

Download Report

the residue theorem • Trig and indefinite integrals

Paperzz.com

Your Paperzz