Question

12. Read Section 88 from the Brown and Churchill Book, 7th edition, to understand the derivation of the most general bilinear transformation that maps the upper halfplane Im (z) >0 in the z-plane onto the unit open disk w< 1 in the w-plane. By imitating the arguments, derive the most general bilinear transformation that maps the right halfplane Re (z) > 0 in the z-plane onto the unit open disk w <1 in the w-plane.

Answer 1

Page. The derisation of the Selutim is vert interesting so ead carefulls Bilinearr 120n fomatin 12 141く1 Relz)o 2-Plane w-Plane Let the neured bilinear transfomatia Wf2 a2t6ad-bc70, et0. iA Ne Knoas that the Bilinear tranafomatiom matp boundarg onto boundaat ie, Re(2)-o (9-ane) 4oer to the e ihele 01 = 1 and the riaht halt plane goes to the inte2i0クz 4the Cincee. Fram 2+6 a a いニ 214 2ニー 汁ond onlà if ω20 Now 22-d if and onla it a00

Page 2 he know thak the mesrse Point the eentre ofa cincee wこ0 Le ince. the aine soe to the eirela いニ00、 are should be 502ニ- and 2ニ-4 Fyod uRs wC mous that thestmmeric Poiht of a paint 2o coith 2espect to d-ands is-2 Since 2- and2- Point aith respeck to d-anes Let 2o be and Paint ram ight half Plone (Ret2)0). Lt -2o then-d-2 aru 8ymmeforiz a 2-20 a 2 t 20 Now we tihd the walue & he Kmous that boneant goes to bondart i.es and Paht 0n anis 90e3 to ang Point onthe boundarナ ot the eincee 1a1に1. Le, the Bieinear tromsfomatin a and c 2z thantfoms on ant Poin&ad-0 toa boundart 叶

Pazes the eisrcee iw11 O20 1 0 +70 X 1al:1 Putting inre t 2-20 1 2720 Ne inow tnat all the Points f the ight half Peane (Rere) 0) goes to the interior sf the eincee (wk1) If 2 2o then w e w0 ahich is the integior Cf the eircle Plane Rel2/2o goes to thi intaidn fthe cincee inw1く1 &オ the Bilinear 90nttrmatim Here 20 mut be ehoosen 19m Ritht halt Plane Rec2)02 plone ie, Rel2o)20

Hence the euired bin bilinear transformatiom is to/2-20 w=fer)= Re(010 Rel2)0 2-peane Hence the Solukfon iA Complete Smmetic Point: The Aimmtri Point ot a pont with epeet to -as is The 8ymmetri Point ta point 7o ith 2espect to danis is-20

Pax- Inverse. Pomt a cincee with centre o Let ws to Ke with nadiu 92. 1et us teKe two PointA Aand B on the 9rar 8uch that OA 03922 then A is called invease Point f B and BiA calted invese Point fA So Imvesse Point叶 entaeof a cirele is the. Point at irtinitt ass DA Here for line we use smmetouc Pomt and forr ciクele we use Iesse Point. Tn the Pobeem, ine g0ed to eincee the bilinear 1andfeamatin HCner symmetrie Poht oes t the Tmvesse Point b the bilinear transformatim 2- ttwD Point Point THANK YOU