WI 1 H H '> II M MW A STUDY OF CERTAIN LOCI A;S8CC§ATEE' MTH ANALYTIC FUNCTIONS OF CCMPLEX VARIABLE Thesis for the Degree of M. 5- Meta M. Ewing I 9 Z 7 T vv‘JWmLNF . ;\ 1- ll: - A. | t . , ‘.‘. . .I n o .l‘ ) 1 .le...‘ .. L 1-: . . .7 «I p .2, r31 . r wk‘.1nnfio‘ I ‘ ' \ Stat. , LIBRARY Uni Midl' wonky ‘ ‘ . MSU LIBRARIES “ RETURNING MATERIALS: P1ace in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. I [*\ Iulounllz iffhfi‘g ‘1'. m 3...; .ca 1.. .. .fllflfluU‘ Al. KL .; I I5.‘ A STUDY OF CERTAIN LOCI ASSOCIATED WITH ANALYTIC FUNCTIONS OF A COMPLEX VARIABLE-- A Thesis Submitted to the Faculty of MICHIGAN STATE COLLTGE of AGRICULTURE AND APPLIED SCIENCE In Partial Fulfillment of the Requirements for the Degree of Master of Science by l r. . Meta M- EWlng 1927 A STUDY OF CERTAIN LOCI ASSOCIATED WITH ANALYTIO FUNCTIONS OF A COMPLEX VARIABLE INTRODUCTION The aim of this thesis is to study nets whose curves in the 2 plane are the maps by means of an analytic function z=f(w) of the lines u=constant, v=constant in the plane of a complex variable w. We find all the nets whose Laplace transforms also enjoy the property of being maps of the lines u=constant, v=constant by means of an analytic function of w. We shall discuss the curves corresponding by duality to the curves of the net under the same restrictions- lo AN ADMISSIBLE SET OF DEFINING DIFFERENTIAL EQUATIONS FOR z=f(w) INTEGRABILITY CONDITIONS Let z=f(w) be an analytic function of a complex variable. If z=x+iy, and w=u+iv, we have xu=yv and xv=-yu. The homo- geneous coordinates1 of a point in the 2 plane (0 I: lflqw, (:7: y(u,v)) 2.1,, are solutions of differential eauations of the form2 C?) 6;“:dé,,,+sav+cej awtdz€p+ékv+ae, PVV : ale)“ +6I’6v #C (9. 1. Define a Net. E- Jo Wilczynski, One Parameter Families and Nets of Plane Curves, Transactions of American mathematical Society, Vol. 12 (lgllfj’p.473. 2. E. J. Wilczynski, One Parameter Families and Nets of Plane Curves, Transactions of American Mathematical Society, Vol. 12 (1911), p. 474. 113C? CH3 Substituting the functions (1) into the first of equations (2) we find that (3) qufalu+6ZV+CXJ yuKTQyu+éyy+CC7, ‘Zua :LLZ“ +6£v +C2. Solving for a, b and c, we find: (4) fl/ 1‘ 1.14.147“ .“é/aekly , 1: 4x?” A: 1:11 fligfiluw K..-) ICE—+1: 5L C : O . Similarly we find the following: I i, (5) (L : WK—IVJ 62' : .XKVSLV_’_ LTKK-AFZ J X5 +er" u 7f 1“. +XV I // A ‘I Ak'gfia‘f‘ {your in, A 3 14 Z VA: “tinder“!!- , X“ + Xv X14. K:- M / u C : 0) C :0 by using the second and third equations in(2). Therefore We can say that the homogeneous coordinates in (l) are solutions of differential equations of the form éuu : {/(fa +56, +06, (72,. V : 017624 +5ch +c'e, 67w : ad's“ + 6’wa + Ca, for the values of a, b, c, a', b', c', a", b" and 0" given in (4) and (5) when x and y are not constants. Let us consider the known relations: (6/ thy") 1V2- “1 Differentiating each of the ecuaticns in (S) with respect u, then with respect to v, we obtain the ecuations: (7/ XMM:§,HV’ ’ruvTcdi/y, Zav:--é/uc4,, KW: *guv. Using these values of these derivatives as defined by the differential eouations (X) (Qua :4'4 19: +45» , guy 261,5“ +5 6V) 6?“, : 4:19.. +56” ) we find that , , an. v-exy war/o, +51.“ whence from (6) we obtain mu. 7‘ng : -431. + e’m, 01‘ (d—g71u#%2¢471 E 1 ita X,: » therefore ” 0' ( " , ’ ” ("j 4? 1‘: é’) ér—CCI. In a similar manner we find / ll ’ I (1, : é and/I/ é : —a", , Hence , fl ’ fl {5/} M:é:—& , ét-él 3‘ . we shall call a set of equations of the form (2) an admissible .ggg of defining differential equations if they admit x, y, 1, such that xu=yv, and xv=-yu. In order that the system of differential eeuations (8) have a solution, it is necessary and sufficient that the following integrability conditions3 be satisfied: [/(IZ + 61,, : wlléi+ drip) 46' +45% a= oc'é +61%; J AZ"; a'Z'+ a}, : dd‘I‘f 4’5”+a'£. , {/0} II I/ flIé/vL éy:a é+é£bo 3. E. J. Wilczynski, One Parameter Families and :31g of Plane Curves, Transactions of American Mathematical Society, Vol. 12 (1911), p. 474. Since a=b'=—a" and b=—a'=—b", these conditions become -flé+fly=—“é”@J 422- é2+ 6V: —éz+a‘ 2-;de 52~6Z2_ é, : -dZ—féz—mw) -45 +41” :—dé-64¢ . These four conditions may be reduced to the following two conditions: . 0d ‘Qv+é“:01 d“ —éy:0. These conditions restrict a and b to be functions satisfying the Cauchy-Riemann differential eouations. The results thus far obtained may be stated in the theorem --Ag admissible set 9: defining differential ecuations for z=f(w), where xu=yv 229.xv=‘yu i§_ ()2) a“: (2‘9“ +A§VJ éuvt-é6“+fley ’ 61y:~.416L—4465J where a gag b satisfy the Cauchy-Riemann conditions- The transformations 4?: 9504), Raw), leave the curves u= constant, v=constant invariant but change the function z=f(w) into E=f(i)- Let us investigate the nature of the function 9 and y>, such that f(fib is analytic. By partial differentiation we find: i » ’7» r , 17 _ l 3 .ill:=§1l aéair-lfi-’ :iflIIELZ he ‘SJ1 0 ) a '3 31 l 35 ‘W' 4” fly ) ”P l - 2/1 5.0;” = ’1." 3 ”3—: : 9'1 014-1-1- : 51": ifif' @V £5 a? (DE 7”‘ 4“ ¢u Since by assumption X§:y; ) X;:—f/J) Xueyy and 1, 2-57“ we find from equations in (13) that 04/ 4b = m- But O is a function of u alone andyz’is a function of v alone, whence from (14) 05'} ¢:K:Kt(+l<,’ ¢2 l7: //\V+K2, Let us consider now the transformation {/é) i=6Ln1 +412 y, Differentiating equations (16) and using conditions (6) we find: n (Ix/”£1! éV+fl/276(=-0421é/“+422f/y3 .. “/11 y“ 7‘g éz’Il <7‘/: — (2/1/77 — 4.22 ykj therefore L {/7) bl” :(LJJ ) (é/Z :va'ZI ' By substituting the values for 822 and 821 found in (17k in (15) and differentiating, it can readily be verified thst the Cauchy-Riemann conditions are satisfied for i and'?. Therefore the most general affine transformation transforming the analytic function z=f(w) into the analytic function i=f(W) i§_3§_affine similarity»transformation. 2- EFINING DIFFERENTIAL EQUATIONS OF THE LAPLACEAN TRANSFORRS The first end minus first Laplacean transforms of the given net are defined in homogeneous coordinates4 by (/y/ 21,.— x,—[¢’x , ISL-4’1) XJIyy‘éCZA/J l'Iyu—éy’ X3: ZV~6<333 E,:Zu—é’z‘7) respectively. In non-homogeneous coordinates they are — - y“ "X ~ _ _ _’ {/7/ X2. “ 1:46" _ , K. - lfzfiéul ) 12 r .C #4,? 1,: .iiu_:él ) (f3, ”1 6—6 4. E. J. Wilczynski, One Parsmster Families and Net: of Plane Curves, Transactions ongmericon—Fethemsticsl Society, Vol. 12 (1911). p. 486. since in our problem 2:1. Substituting the values of In )1; J (7; , ($7,;— in (13) into the second equation of (2) we have a I l I Kéfi924K4QZ+Afl§V+CQ 0r 6‘7; t 2%; 6);? +/€_ (9‘7 +f__ @- ‘/ ’ ‘/ .r \/ / Lat 4 :Z J g :£, , C T.C :‘0 /4’ K K2 ’ The eXpressions (18) become \ I —5’z:/4/(——/ ;:A6“‘fyewd(éu+aV~MA ‘ A — V 7::K 9““ 145m (éu167V“47)1 15. where K and M are arbitregg_constants are the most general solutions of the case i2_which equations (2), (18) and (19) are all germissible and such that the solutions are real and satis§1_the Cauchx—Riemann differential eouations. 4. DEFINIE 3G DIFFERE I’TIAL EQUATIONS OF THE TAHGENTS TO cu AND 0v The tangents to v=constant are i/7w/‘//—% j-r/(z/M—Auyfl— 0. Let Ll) L2 and 1/ be the Plucker coordinates of this line. Then La 95’ ((1: L , LA- 1% , /f/. , W¥ l‘u-Xu Differentiating; he ecuetions (33) end making use of ecuetions (3) We obtain the following; ’ {W ' (a) “'1 7’; : ‘ <6 V ' u v a "A“ (34/ j A %«/&{+ {m LdL! , (j v 4% KIA“): X” We 4%? +4va ’ W < Again aloh4:;é7/3W:»d JWJ6€W3“CWI ~241Vfi3¢2jfi:)1 HM ”44 A’“;dg/z“1.,2m“;_/“’jzy}+ “fie (fibflhu,:;cfld(hfica M.,u” 2¢51“+1“1“M—41VIZ2XZY“+44X”“)9" 711(A'AM/1t'h/ vim/2A (”.2 rag/l “Cid juiiwjérv} -.'~/1W (if/“W _— g (A luff/{“9 a2/1“’+AW(W-,25/iw ‘“’—2 AWE) y“ +(a2/é/1Wz -a .,u/\( +241mI/ilul)/ }2—5 AWE: ) («I 1 wherein A 2:, l “/. 16. Substituting the first derivativec othined in (34——a,b,c,d) in equations in (8), we find: (35) (a) 2/“. : {a ox, X“: a, 2:”- 44:12? zit/ll u__ +9: 613(u':_/<fl_ 3:21.14 ) (6%, “ ,,. v 2 log-4 a, Well? :igf'flfl’fl’i’)gu+(— e34 {1%.329— J > 1/ ' J u (“I 4 N I“) I! (“/) II (N) / " ‘11) «j“n,:—wuxAE¢a;3u+éOlif%§_ L Hu+kéc&/1-agjj (V. ' _.___..--- 4&31 Eliminating .X“ and./lv between the equations formed by setting the equivalents of corresponding second derivatives equal in (34) and (35) the coefficients of the defining differential equations of the tangents to (2‘ are (a) (36/ w 6%, : 34 + 5% — gale , (14/ ,3 .— -4 é 7(a) / : . (“l l (U) — - y ..... a"! (I ""1 '3 . - u. (C) CY, _ é +_2/__ A“: J ( //.J7 ((16! ) I 4 A ,, (“l , l _. // ,. _ .- - u w’“ ‘4:@“+“4/ ’ Qfioi‘ éféé- In a similer manner we find the defining differentiel ecuations of the tangents to (1V are _ _ (V) _ (fU M/q273dfw%L-J/$k , @042:_4J / V I (C/ 0(2 : -—é 7Lét: _. /1(': , (d)/’32_/': 4‘ A(: 6 W)- H / ) ) // II {V} " /‘ /"7 - —‘- —~ V . (1.2 %_- /é My f 04/ , (y 2/91 , 54 2rd Let us impose the condition that the defining differential equations of the tangents to Cste admissible- ' I r )2 If ()(I: f” we have ;?a +‘éfi.__ 6, 01‘ (“I (q) 3/8 + é; —- =2 A“ : a— w 6 Au) ) then 1(a) . i ‘ _ A “F1607“ “ ‘24 2““ , also. / (u) z(—/“ __a,_ta : Z“ ' Hence (a) {1L ‘ 247‘— fig AW 6 ‘ therefore Ja — é} :2/1+ 4‘, £3 _ J 01? .15 + £44 : 0, 6 6 hence If /9/ I - (Y! T//_)}, we have , (‘41 , (u/ bh_é_ + “AV : 3(7 +£_/l_k_. 6 X14) Am; whence , . (u) (”j O‘céz 4. _/\V : 35+ 9/11, é KID" AM) 01‘ (w A V “ “ 2 g _ .41: “AW ’ also (“I 3éde514J: —..é, A“ 01‘ (04/ _ /\ V -— - .26 ‘Al‘f.’ ’ therefore 17. 2-4, 18. and 41,20‘ 07'.é is a function of u alone. But we know from theorem I that 6(“:4,) Then a?u~:¢9) or an is a function of v alone, but 41V:_.Au— , therefore a and b are constants. Under these conditions the coefficients (36) become (32/ Xe. fit :_A' , «31.52) (X’:él /3,/—_*av’ fltg, In a similar manner we find (37” 0213-4, /3::- A) (1:1:42’ (2:36) /31:—4, ”2:4. By substituting the values for ,2 and ii of (37) in equivalents Of/LL and 1) in (38) we have 44— 4 /0L : 61/(C V Mféqjean— __/z/)/—/-_ éA ed "gym/ewueray m)_ KJe a“ 6yt’0Mééumy—Mj é/u'm(6u+anr~ W) “9"" ‘ZzZfléwu-meéwm ”'1’ MA 7,): /1’ {46“ w/3(6u+4y~M)~A e“ H ' _iéfii‘iwfllf—r Kxeau 6V(AC«/>(6u+¢v—/V/j+d m;(éuMV—Wjj~ylréa“ 6 (am/”NV ”0‘64“”(6umu ”0) 01‘ m év K Qau [21/36ch éX/Cwo (64*M— /1/7j— (d X‘l’6Q/W (Au +du— M)f) / 2": __/_:G él/Zd‘ (76% (AU+dV 47/ é/Mw (éu+62V—~ “W; K G“ :;§(bx aj)€wo (5a+ay- Mj+ (UK-1‘ 6%)Qm(éu+au Ivyf 3 19. Let ¢.:é¢/+4y-/y/ and (QTfl/J‘éy) then _ a , , (40} /( : E (”.34-”)«1 (75 + A Capo 7(5)) EAK' ‘ V: _d:6 ’ -6( L',(L-~J¢+ A/JILM _, . ‘6/\ K 97$) We may readily verify that 24 : /4q/ andzdwe—frn and that /A, and 2/ satisfy the system fiuur—Czeafiéfivy (9“,,2—écQR—~a(9y J eyy aQIA +é9y '. ‘ I We may summarize our results as follows: The most general net whose defining_differential equations are admissible and the defining differential equations g:_whose minus first Laylace transform is also admissible is defined Ex _M -4 X : 14, 8”“ V can (éu +au «4/0, au‘AV . (17/375 6 gum (6a+av—/I/lj, wherein E? /M at ¢§ are arbitrary constants. J J y ) Any other net with these properties is :_similar affine transformation gf_any given one. The admissible defining differential enuations are: qut‘éeu. +deV) e -. —a eu-gev, VV It follows that the defining differential equations Q£_the first Laplace transform is also admissible._ The defining equations gf'both Laplace transforms coincide with the defining equations Qf_the original net.’ Hence the Laplace transforms are similar affine transformations of the original net. Again the defininquifferential equations of the nets corresponding §1_dualitz t__the curves of the original net are admissible if and only if a and b are constants. Each inthe net§_thg§_correspondingpgl duality to the curves of the net i§_§_similar affine transformation of the other. The defining equations of the dual nets are: [Quél:-é? (gt/(“69V’ aw : éau _ as,” (El/y ’: flea—féfly o THE SUBJECT OF THIS THESIS WAS SUGGESTED BY DR. V. a. GROVE. IF THIS PAPER POSSESSES AHY MERITS, THEY ARE, DUE. TO THE INSPIRATION AHD THE HELP - FUL SUGGESTIONS GIUBH BY HIM. TD THE I WRITER WHILE DBUBLOPIHG THE. 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