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, _l manmitt; Jul2.15.5“ . {xi.u;.:uﬁsm.. (~34ng .ggl,,,,.,_‘;u=~ “ivaim» 1,"... Chapter 2 VARIABLES AND FUNCTIONS A symbol, suchas z,'which can stand for any one of a set of complex numbers is  called a samples: variable. . If to each value '_ which a Complex variable 2 can assume there corresponds one or ' .mOre values of a complex variable 10, we say that w is a function of _z and write w =. f‘(z)
' Z'or 'w: 6(2), etc; The variable 2 is sometimescalled an independent amiable, while w is called a dependent Wﬁdble The value of a fﬂﬂCtion' at .2 2a. is often written ﬂu). Thus
if N?) 7—: 22, then 'f(2£) = (23')2 2 "4,  ' . 4 ._ .. I SINGLE AND 'MULTIPLEVALUED FUNCTIONS If only one value of 3w corresponds to each value of 2, We say' that w'is a singlewulned ' ‘ function of z or that ﬂz) is single~v‘alued. If/more than one value of 7w corresponds to each value of 2, we saygtliat wiisj'a multiplevalued or manyfualned, function of z. _
' ,‘A'multiple—valued function can be considered as a collectionlof singlevalued functiOns', , I each member of which is called a branch of? the function. 'It is customary to'consider. one ' ' particularfmember'asaxprinc'ipal branch of themulti'pleavalued function and the value of "  theiunction ”correspondingi 'tojthis' branch as th'e‘pninc'ipql’nblne; ' \ Examine 1; _ If to = 2”, than to each value of 2 there isonly uneavalue‘ of w. Hence 150F112) = 23‘ is a  '
x : ' singlevaluediﬁunction of z. ‘ ' , " ' ‘ ‘ A __ ‘ amultipledvalued (inthis case twovalued) tunction'ofla. ' , _ _ _ _ .
' Whenever we speak of function we shall], unless otheerise stated, assume singlerimmed ‘ Egan113192:  A"I:II toga/2, theniito‘each value of z thererare Novelties of 10} 'Hence w: f(z) =_'z'1/5_3‘is INVERSE'FUNCTNNS' , * _.  s ' n~ 1 ‘  ' 1 ‘
, If n) = f(z), then we can aISQ consider z as a function of w, written 2 = g(w) = f" (w).
The function f“ is'roften called the inverse function corresponding to f._ Thus w = f(z) and w =‘_f‘1(z)r_are inverse functions of each other. ‘ TRANSFéRMATIONS  If w :15 +131; (where stand a are real) is a singleevalued function of z ':o:r+iy (where ' I ' sand 3; are real),rwercan write n+2’v ’2 f{_x+iy). Byequatingreal and imaginaryparts ' this is seen to beequiv’alent to Thus given a point (any) in then' plane, such as P in Fig, 21 below, there corresponds a
point (n, 1'2) ‘in the to plane, say P’ in Fig.. 2—2 below.."The 'set of equations ,(1) [or the ,
equivalent, w : f(z)} is calleda transformation. 'We Say that point P is mapped or ~ '
transfonned into point P’V'bygmeansof the transformation and call .P’ the imageof ‘P. 33‘ 34'. ' . FUNCTIONS, LIMITS AND CONTINUI‘I‘Y {crime 2 Example: If to = 22, then u + 2'1: 2 (2: +139)? 2 22 — y: + 259:?) and. the transfoﬁnation is
u : 22—y2, o : 23y. The image‘ of a point (1,2) in the 2 plane is the point (—3,4),in
the in plane. ' ' Fig 21 . ' _ Fig. 22 Ingeneral, under a transformation, a set of points such asthose on curve P6) of
Fig. 21 is mapped into a correspbnding set of points, called the image, such as those on
curve P’Q’ in Fig. 2—2. The particular'characteristics cf the image depend of course on
the type of function re), which is sometimes called a mapping function. If/f(z.) is multiple
valued, a point (er'curve) in the 2 plane is mapped in generalinto more than one point
(or curve) in the or; plane. ' CURVILINEAR COORDINATES Giventhe transformation w = Hz.) or, equivalently, u 2 u(x,y), o i okay), we call
(x, y) the rectangular coordinates corresponding to a point P in the 2 plane and (u, o) the , curvilinear coordinates of P. 2 plane ' w plane Fig.23 ' t 1 Fig. 274 The curves u(a:,y) = 0;, 142:, y) 2 c2, where c: and 62 are constants, are called
coordinate curves [see Fig. 23] and each pair of these curves intersects in a point. These
curves map into mutually orthogonal lines in the ‘w plane {see Fig. 24]; ' THE ELEMENTARY FUNCTIONS
1. Polynomial Functions are deﬁned by .w : aoz" + (112“"1 + ' ii + (tn—12 + an = 13(3) (2) Where do aé 0, a1, . , .1, on are complex constants and n is a positive integer called the
degree of the polynomial P(z). ' ' The transformation on = az +1) is called a linear transformation. J
i
CHAP. 2]  r _FUNCTIONS, LIMITS AND CONTINUITY 357 2. Rational Algebraic Functions are rdeﬁnedby, ' P_(z) ‘ ,
a .   2’ 0(2) ( 1
WhereIP(z) and 9(2) are polynomials. We sometimes call (3) a rational transfermation 112 + 1)
oz + d
.tional linear transfdrmation. , The special case w I‘— where ad— be aé 0 is often called a bilinear or frac 73. EXponential Functions are deﬁned by w — —ez : 32“” 2 eicosy—Fisiny) I I (4)
where e— — 2. 7I1828.. . is the natural base of logarithms. If a is realiand positive, we
deﬁne 7. _ 1.  1 . . I a: ._.. 8: na . . (5) ' Where In a is the natural logarithm of a. This reduces to (11) if a— — 9.
Complex exponential functions have properties similar to those of real exponential
functions. For example, 62:  322— — ezr+2=, 321/32?— — 32122_  ‘4. Trigonometric Functions. We deﬁne the trigonometric or circular functions sinz,
cosz, etc” in terms of exponential functions as follows. sinz— (HM—e ‘ cosz—eizFe—i'z
, ‘ 2i 1 _ 2..
1 '  secz— 1  2 cscz—i— 2% _‘ cosz _ 32+?“ ' . sinz 9273'?“  sinz 'eiz—e—‘2 cos'z , “2+?“
stanz;'——=sz—'—2; cotz= 3 =L;—_£T)
‘  cosz 2(3 +3“ )' , slnz ‘ e ——e‘ Many of the properties familiar in the case of real trigonOmetricI functions also _
hold for the complex trigonometric functions. For example, we have ‘ sin2z + cos2z = 1 1 +tan2z 2 'sec2z  1 + cot2z— — csc2z
sin (2): — sin 2 ' cos (#2) =' cos 2: tan (—z)‘= ~— tan '3 ’
" sin (2;: 22) = sin 21 cos I22 + cos 2; sin 22 t '
cos (2; if 22) m I _cos 21 _cos 22 '1— sin 2; sin 22
"tan21i'tan22 ' ' 7
, tan (21 1 22); =, ——’——— 1 I tan 21 tan 22 ' 5; Hyperbolic Functions are deﬁned as followsr . e2 —_e‘2  . e“ + 6'2
smh Z = , ~—— ' 1 cosh z : ___...
J ' . = '. 2 _i I . 1 2 I
‘ . _ 1 _ 2 _ ' , __ 1 _ _ 2'
secha ___ coshz _ W cschz _ sinhz —' 322—972
tanh z _ sinhz _ 9’ ‘73" (:0ch _ coshz ___ 62+?“ coshz e2 + e"; I ' sinhz _e.‘ —‘ 3'3" ' The following properties hoitLI cosh2z —' sinh2z z_= 1 1 —— tanh2z # se‘chl2z‘ coth2z — 1 =7csch2z
sinh (—2): —' sinh z cosh (—z) — cosh z. tanh (~12): ,._ tanh z
sinh (211 22) = sinh z: cosh 22+ _ Cosh 'z; sinh— 22 '
'I cosh (z; i. 22)  = cOsh zi cosh 22 i sinh a; Vainh 2:2
tanh (21 i 22) = , tanh Z] i tanh 22 V . 1 it tanh z: tanh 2'2 It 36 FUNCTIONS, LIMITS AND CONTIINUITY [0111113. 2 The following relations exist between the trigonometric or circular functions and the hyperbolic functions:
sin iz .2 i sinh 2 cos iz = cosh zr tan is i tank 2 sinh is = i sin; cosh it cos 2‘ I tanh is _= i tan 2 ll 6. Logarithmic Functions. If 2 = c”, then we write in 7 ln‘a, Icalled the natural toga,
rithm of 2. Thus the natural logarithmic function is the inverse of the exponential
function and can be deﬁned by _ I w 2 Inc» = lnr+i(0+2kw) kEO,+1, +2,. where z— — rema — WWW“), Note that 1112 is a multiplevalued (in this case inﬁnitely . manyvalued) function The principalvalue or principal branch of lnz is sometimes
deﬁned. as lnr + it) where 0_ S 6 < 211'. However, any other interval of length 271'
‘can be used, e. g. —nI' < 6_< ... 71', etc.  I The logarithmic function can be deﬁned for real bases other than e. Thus if
2:51“, then wzlognz where c>0 and 11,950, 1. In this case z=e"’hm and so
10 —— (1n z)/(ln 0,). ‘ , 7. Inverise Trigonometric Functions. If z—  sinw, then 10— — sin 1,2 is called the inverse
sine of z or are sine cf 2. Similarly we deﬁne other inverse trigonometric or circular
functions eds 1z,tan"12:, etc. These functions, which are. multiplevalued, can be expressed in terms of natural logarithms as follows. In all cases we omit an additive I constant kai, k: 0, +1, +2,. ., in the logarithm.
sin'lz = —.Iln (iz+1/1— 22) Icsc"1z “If”: ilanl/zE—l) see“‘z lln(1+‘f:?) 3 1 1+iz I 1 2+2?
1 —. 7 —]_ .
tan z —2i in (12.2) out a —.2?’ 111(z _ i) 8. Inverse HYPerbolic Functions. If z :sinhw then 20 = sinh—lz is called the inverse I hyperbolic sine of 2. Similarly we deﬁne other inverse hyperbolic functions cosh 12;,
tanh 12, etc. These functions, which are multiplevalued, can be expressed in terms
of natural logarithms as follows. In all cases we omit an additive constant 219.5, k: 0, +1, +2,. . ., in the logarithm.
I sinh‘I z = 1n (2 + 1/zz+ 1) csch’l 2 In (L ”22“) cosh"2 = 1n(Z+\/2"1) seth—lz = m(_1+ 1—22)
2
_ 1 1+2 , I 1 2+1
1 —— .. — _ _
tanh z —,21n(1_z) coth lz __ 2]n(z_1) 9: The Function 2", where a: may be complex, is deﬁned as 8““ 2. Similarly if f(z) and 9(a)
are two given functions of a, we can deﬁne 112)”) = eg‘z’inf‘z’. In general such func
tions are multiplevalued. 10. Algebraic and Transcendental Functions. If it: is a solution of the polynomial equation Po (2)112" + P; (2)21)“1 +   ' + Pu—1(z)w + Fri (2) = OI (6) where Po $ 0, 131(2), .. ,P (z) are polynomials in z. and n is a positive integer, then
in: f(z) is called an algebraic function of 2. Example: 111*“ — 2W is a solution of the equation wz—z = G and so is an algebraic function of z. ' .33 . V FUNCTIONS, LIMITS (AND CONTINUITY ' '  {CHAR 2 is the limit of ﬂat} as 2 approaches Zn and write lim f(z): _ if for any positive num '
2DZ“ ber e (however small) we can ﬁnd some positive number 8 (usually depending on e) such
that f(z)—l < 6 whenever 0 < 2—Zo[ < 8. In such case we also say that f(z) approaches 1 as 2' approaches an and Write f(z) >l
as a '> 20. The limit must be independent of the manner in which it approaches 20. Geometrically, if 20 is a point in the complex plane, then lim f(z) it if the difference
2—92“ in absolute value between f(z) and t can be made as small as we wish by choosing points z
suﬂ‘iciently close to 20 (excluding z: .20 itself). '  ' 2 5e ' . . .
Example: , Let Hz) =. {3 :n :3. Then as 2 gets closer to 1' (i.e_.a approaches it), f(z) gets closer ' to £2: —1. We thus suspect that lim ﬂz) = —1. To prove this we must see whether zbi the above deﬁnition of limit is satisﬁed. For this proof see Problem 23. . ,
Note that lim f(z) at ﬁt), i e. the 'Iimit of {(2) as 2 * i is not the same as the value z_+i of ﬂz) at zi—“i, since f(t')= 0 by deﬁnition. The limit would in fact be 1 even if f(z)
were not deﬁned at z=i. , 7 _ '1 When the limit of a functionexists it is unique, i.'e. it is the only one.(see Problem 26);
If f(z) is (multiplevalued, the limit as 2 +20 may depend on the particular branch. _ THEOREMS 0N LIMITS 7
I If. lim f(z) z A and lim 9(2) = B, then zobzo , z+zo
1, 1im{f(z.)+9(2)} = lim f(2) + lim 9(2) =‘ A +'B —
. 2"30 zan erzo
, 72 351: {f(Z)—9(Z)} = hm f(z) — lim 9(2) = A _ B
' 3_. 133° {sages = {5333(2)}{132 9(2)} = A3
lim f('z) 
f(z) .: zuz. _ A . _
4lilgng(z) 3129M.— E 1: BTU .
INFINITY . By means of the transformation 10— — 1/2 the point 220 (i. e. the origin) is mapped
into ur — 00, called the point at inﬁnity in the to plane. Similarly we denote by z— — no the.
pointet inﬁnity in the 2 plane. To consider the beha’vior'of ﬂat). at z = no, it sufﬁces to let '
z: ' 1/w_ and examine the behavior of f(1/w) at 1020. ' We say that lim f(z)— —— l or f(z) approaches 1 as z approaches inﬁnity, if for any 2—»: e) 0 we can ﬁnd M > 0 such that [f(z) — ll < 2 whenever z] > M. We say that lim f(z)— on or f(z) approaches inﬁnity as a approaches 29. if for any ZDzo 'N>0 we can ﬁnd 8>0 such that f(z)( > N whenever 0 < [z— 20} < 8. ' CONTINUITY Let f(z) be deﬁned and singlevalued in a neighborhood of 2220 as well as at 2:20
(Le. in a 8 neighborhood of 20). The function f(z) is said to be continuous at z=zo if Iim f(z) : f(Zo)_. Note that this implies three conditions which must be met in order that
2H2“ _ "
‘ f(z) be continuous at z: 20: ' h'm'iserwt'qlﬁ'ira A slowaw“ 'aai‘fz 1'7? altim "mien: .. :‘ ' ‘ point a.) such that f(z
‘7 on elbutnot onthe particular point ao,We say that ﬂz) is uniformly continuous inthe region. CHAP.‘ 2] ‘ _' FUNCTIONS. LIMITS AND CONTINUITY ' . 39 1.' lim f(z} = l mustexist '2. flag) must exist, i.e. 7(a) is deﬁned at at
3.. l=f(Zo) ' . v ' I Equivalentlﬁif ﬂz) is continuous at 2.} we can write this in the suggestive form lim f(z)i = f(lim z) . ‘ , . ‘ . zozo z—rzo ‘ 2 a;  .7  , 
Example. 1: If f(z) = S :=; hen from the Example on Page 33, gig: re) = —1_. But. no = 0. Hence lim ﬁx) 9* f6 and the function is not continuous at z: i.
 , z+£ _ l .
Example 2: ' If f(z) =z2 for all 2, then ling ﬂ?) =7 ﬂ!) = —'1 and ﬁx) is continuous at 2:11
z—r _  . . Points in the z plane where fizj' fails to be continuous are called discontinuities of ﬁzz); and f(z) is said to be discontinuous at'these points. If iim ﬂz) exists but is no equal to f(z.)), we call '20 anemooable discontinuity since by redeﬁning. f(zoi.to be the sanie
as lim7f(z)r thefunction becomes continuous. ' zsrzo Alternative tothe above deﬁnition of continuity, we can deﬁne f(z) as continuous at .
23:20 if for any £>0 we can ﬁnd 8>O such that 1f(z)— f(zo) < 6 Whenever [2—29] <8. '
Note that this is simply the deﬁnition of limit with l=f(zo) and removal of the restriction
thatZSéZo. ‘ I I 'I ' ' 5 7 To exainine the continuity of 112) at z = on, we place 2: llw and examine the continuity ' .of ﬁllw) at w=0. ' . 
CONTiNUITY IN A REGION .  A functifm ﬂz) is said to be continuous inn region if it is continuous at all points of
the regiOn. . . _ _ _ _ THEOREMS on CONTIﬁUITY _ i 7 u , _ . _
‘ ' Theorem L If f(z) and 9(2). are continuous at 23:20, so also are the functions '.,lf‘(z)'+ 9(2)? ﬁg) 901), 1139(2) and. iii—i, the last only if.g(zo)_ a? 0_. VuSimilar results hold for
"iqsiinuity in a resibm 7 ' ., ' ' f I 9' I ' Theorem 2; Aniongthe functions centinuous in every 'ﬁn'ite region are (a) all poly _ 'nomials, ”(b) 32, (c) sin 2 andcos z,‘ i " Theorem 3. If w¥ f(z) is continuous at 2:20 and dz=g{§) ‘is continuous at 22;];
andital": 9030); thenjthe function w 5: {[96]}, called a function of a function or, composite
function, is continuous at C :41}. This is simetimes briefly stated as: ' A continuous function  of a continuous function is continuous. ' . Theorem 4. If f(z) is continuousin a closed region, it is bounded in the region; i.e. _
there‘exists a constant M such that f(z)] < M for all points z of the regidn. ', , “Theoremx5. If ﬂz) is Continuous in a region, then the real and'inlag'inary parts of
ﬁn} arealso continuOus in the, region.  ‘ " UNIFORM CONTINUITY _ ; . _ :
' ' Let Hz)" be continuous in a region. Then by deﬁnition at each point Zn. of the region
andforany'e > 0, we can ﬁnd 3' > 0 (which will in general depend on both 6 and’the particular _ ):— f(zo)[ < {whenever [szd < 8. If We can ﬁnd 8 depending ...
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