Complex Numbers. Prepared by: Prof. Sunil Department of Mathematics NIT Hamirpur (HP)

Transcription

1 th Topc Compl Nmbrs Hyprbolc fctos ad Ivrs hyprbolc fctos, Rlato btw hyprbolc ad crclar fctos, Formla of hyprbolc fctos, Ivrs hyprbolc fctos Prpard by: Prof Sl Dpartmt of Mathmatcs NIT Hamrpr (HP) Hyprbolc Fctos: I mathmatcs, th hyprbolc fctos ar aas of th ordary trgoomtrc, or crclar, fctos Th basc hyprbolc fctos ar th hyprbolc s "sh", ad th hyprbolc cos "", from whch ar drvd th hyprbolc tagt "tah", aay to th drvd trgoomtrc fctos Th vrs hyprbolc fctos ar th ara hyprbolc s "arsh" (also calld "ash", or somtms by th msomr of "arcsh") ad so o Jst as th pots (cos t, s t) form a crcl wth a t rads, th pots ( t, sh t) form th rght half of th qlatral hyprbola Hyprbolc fctos occr th soltos of som mportat lar dffrtal qatos, for ampl th qato dfg a catary, ad Laplac's qato Cartsa coordats Th lattr s mportat may aras of physcs, cldg lctromagtc thory, hat trasfr, fld dyamcs, ad spcal rlatvty Th hyprbolc fctos tak ral vals for ral argmt calld a hyprbolc agl I compl aalyss, thy ar smply ratoal fctos of potals, ad so ar mromorphc Thy wr trodcd by 8th ctry Swss mathmatca Joha Hrch Lambrt

2 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: A ray throgh th org trcpts th hyprbola y th pot ( a, sh a), whr a s twc th ara btw th ray ad th - as For pots o th hyprbola blow th -as, th araa s cosdrd gatv (s amatd vrso wth comparso wth th trgoomtrc (crclar) fctos) Dftos: For all vals off, ral or compl () () th qatty th qatty Ths, sh ; as follow: s calld hyprbolc s of ad s wrtt as s calld hyprbolc cos of ad s wrtt as sh, Th othr hyprbolc fctos ar dfd trms of hyprbolc s ad cos sh tah ; sch ; cosch sh Rmarks: sh sh coth sh 0 0 ; 0 ; 0 sh

3 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: sh, ad tah Rlato btw hyprbolc ad crclar fctos: Prov that: () s ( ) sh (v) cosc( ) cosch () cos ( ) (v) sc ( ) sch () ta ( ) csch, sch ad coth tah (v) cot( ) coth Proof: () Sc w kow that cos ; Pttg ths qatos, w gt s

4 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 4 Vst: s ( ) ( ) ( ) ( ) ( ) sh Hc s ( ) sh () cos( ) ( ) ( ) Hc cos ( ) () ( ) ( ) ( ) s sh ta tah cos Hc ( ) tah ta (v) cosc( ) s ( ) sh Hc cosc( ) cosch (v) ( ) ( ) sc sch cos Hc ( ) sch (v) cot( ) sc cos s ( ) ( ) Hc cot( ) coth sh cosch sh coth sh Formla of hyprbolc fctos: (I) Prov that (a) sh, (b) sch tah, (c) coth cosch Proof: (a) For all vals of, cos s Pttg, w gt cos ( ) s ( ) ( ) ( sh ) [ cos ; s( ) sh ] sh [ ] (b) W kow that sh

5 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 5 Vst: Dvdg both sds by, w gt tah sch sch ta (c) W kow that sh Dvdg both sds by sh, w gt coth cosch coth cos ch (II) Prov that (a) sh ( y) sh y ± sh y ±, (b) ( y) y ± sh sh y (c) tah( y) Proof: (a) sh ( ± y) s( ± y) ±, tah ± tah y ± ± tah tah y ( s cosy ± cossy) sh s s sh ( sh y ± sh y) cos sh y ± sh y (b) ( ± y) cos( ± y) [ cos] (c) tah( ± y) cos cosy ssy y sh sh y y ( sh sh y) [ ] y ± sh sh y sh ( ± y) sh y ± sh y ( ± y) y ± sh sh y Dvdg th mrator ad domator by (III) Prov that (a) tah ± tah y ± tah tah y y, w gt tah sh sh, tah (b) sh

6 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: (c) Proof: (a) W kow that Pttg, w gt tah sh, tah tah tah tah s s cos ( ) s( ) cos( ) sh sh s sh sh ta Also s ta Pttg s ( ), w gt ta ta (b) () W kow that Pttg, w gt cos tah ( tah ) cos cos tah sh tah s ( ) cos ( ) s ( ) ( ) ( sh ) coh sh () W kow that cos cos Pttg cos, w gt ( ) cos ( ) Ths also () W kow that Pttg, w gt cos cos s ( ) s ( ) ( sh ) sh Ths also sh (v) Sc w kow that Pttg, w gt ta cos ta tah sh tah

7 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: cos ( ) ta ta (c) W kow that Pttg ta ( ) ( ) ( ), w gt ta ta tah ( tah ) ( tah ) ta ta ta ( ) ( ) tah tah ( tah ) tah tah tah tah tah tah (IV) Prov that (a) sh sh 4sh, (b) 4, (c) tah tah tah tah Proof: (a) W kow that Pttg, w gt s s s 4s ( ) s( ) 4s ( ) sh sh 4( sh ) sh sh 4 sh sh sh 4sh (b) W kow that Pttg cos, w gt cos 4cos ( ) 4cos ( ) cos( ) cos 4 (c) W kow that Pttg tah ta ta ta ta ta ta, w gt ta( ) ta tah ( tah ) ( tah ) ( ) ( ) ( ) tah tah tah tah

8 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 8 Vst: tah tah tah tah (V) Prov that () sh AB sh( A B) sh( A B) (VI) Prov that (), () Ash B sh( A B) sh( A B), () A B ( A B) ( A B), (v) sh Ash B ( A B) ( A B) C D C D sh C sh D sh, C D C D () sh C sh D sh, () (v) C D C D C D, C D Ivrs hyprbolc fctos: Dfto If C sh D C D sh sh, th s calld th hyprbolc s vrs of ad wrtt as sh Smlarly, w ca df, tah, tc Th vrs hyprbolc fctos lk othr vrs fcto ar mlt-vald, bt w shall cosdr thr prcpal vals Rslts: To show that () sh ( ) Proof: () Lt,, () ( ) () sh tah 0, whch s qadratc, th sh ( ), so w gt

9 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 9 Vst: ( 4 4) ± ( ) ± Takg th postv sg oly, w gt ( ) ( ) Hc sh ( ) () Lt, th ( ) Ths bg a qadratc 0,w hav ( 4 4) ± ( ) ± Takg th postv sg oly, w gt ( ) ( ) Hc, ( ) () Lt tah, th tah Applyg compodo ad dvddo, w gt ( ) ( ) Hc tah Now lt s solv som problms: QNo: Elmat from p cos ch qsch r 0, p cos ch q sch r 0 Sol: Gv p cos ch qsch r 0, () p cos ch q sch r 0 () Solvg () ad (), w gt

10 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 0 Vst: cos ch sch qr q r rp r p pq p q qr q r cos ch ad pq p q rp r p sch pq p q Also, sh pq p q rp r p pq p q qr q r ( p q p q) ( qr q r) ( pq p q) ( rp r p) ( rp r p) ( qr q r ) [( p q p q)( qr q r) ] [ ( pq p q) ( qr q r) ]( rp r p ) [( p q p q)( qr q r) ] [ ( pq p q) ( qr q r) ]( pr p r ) As QNo: If Sol: Gv y ta, show that sh y ( ta cot ) y ta y y ta ( ) ( ta ) y ( ) ( ta ) y ta () Th y ta cot From () ad (), w gt y y ( ta cot ) ( ta cot ) sh y Ths complts th proof () QNo: Prov that () ( α β) ( α β) shαshβ Sol: () Sc w kow that, sh () ( α β) ( α β) ( sh α sh β) y y sh y

11 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: αβ ( αβ) αβ ( αβ ) ( α β) ( α β) α β α β α β α α β ( β β ) α ( β β ) β β α α ( )( ) sh αshβ Ths complts th proof β β α α ( )( ) ( αβ) () Sc w kow sh( α β) ad ( α β) sh α β α β α αβ ( ) ( ) β α β α β α β α β αβ αβ αβ αβ [ ( ) ( )] αβ ( α β β α ) 4 α α β Ths complts th proof QNo4: If β ( sh α sh β) ta y taα tahβ ad ta cot α tahβ, prov that ( y ) shβcoscα ta Sol: Gv ta y taα tahβ ad ta cot α tahβ Sc w kow ta( y ) Sbstttg th vals of ta ( y ) ta y ta ta y ta ta y ad ta, w gt taα tahβ cotα tahβ tahβ taα cotα ta αcotα tah β tah β 4 ( ) tahβ ta α sch β ta α sc α shβ β ta α shβ β sh βcos cα s αcos α Ths complts th proof QNo5: Prov that () ( ± sh ) sh, αβ ( αβ)

12 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: () sh tah tah Sol (): Sc w kow that ad sh ( ) sh ± ± Takg postv sg, w gt ( ) sh ( ) sh Takg gatv sg, w gt ( ) sh ( ) sh Hc, ( ) ± ± sh sh Ths complts th proof () Sc w kow that ad sh

13 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: Now sh sh sh sh tah tah HS L Sbstttg for ad sh, w gt ( ) Hc, sh tah tah Ths complts th proof QNo: Eprss trms of hyprbolc ad mltpls of Sol: Sc ( ) ( ) Epadg by Bomal thorm, w gt c c c c c c c c ( ) ( ) ( ) ( ) ( ) ( ) 5 5 As QNo: If tah s, prov that ta sh Sol: Gv tah s sh

14 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 4 Vst: ( ) Also cos s ( ) s ta cos Ths complts th proof QNo8: If Sol:() ( ) sh ta tah, prov that () ta sh, ta ta ta ta () cos, () ta ta ta tah tah ta ta ta h sch π ta 4 sh sh sh Ths complts th proof () To show: cos Sc sh ta sh ta sh ta sc sc () cos cos sc [from ()] Ths complts th proof

15 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 5 Vst: () Gv that ta tah ta Applyg compodo ad dvddo, w gt ta ta / / π ta 4 / / Ths complts th proof QNo9: If Sol: () W hav / / / / π ta 4 / / π ta, prov that () 4 π ta 4 / () / By sg compodo ad dvddo, w gt / / / / () Sc tah Hc QNo0: If ta ta tah π ta 4 ta ta tah tah / / / / tah ta ta ta ad π ta 4 ta ta ta ta ta π π ta ta 4 4 sc, prov that () tah ta, / /

16 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: () π 4 ta Sol: () Gv that cos sc ta ta tah tah ta ta,cos tah tah Applyg compodo ad dvddo, w gt ta tah ta tah Ths complts th proof () Sc w kow ta tah Th, w hav tah ta / / / / ta Applyg compodo ad dvddo, w hav / / 4 ta ta ta π π 4 ta Ths complts th proof QNo: Show that ta Sol: Show that ta ta Now RHS / / tah ta / / / / / / / /

17 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: / / Ths complts th proof / / QNo: Prov that () sh ( ) Sol:() Lt sh () tah tah sh ( ) ( ) cos ch ( ) sh () () From () ad (), w hav sh () sh Now, tah (v) sh tah From () ad (v), w hav sh Also, tah cosch cos ch sh Hc, cosch From () ad (v), w hav sh tah sh tah tah (v) cosch cos ch sh sh sh ( ) cos ch cosch cosch sh (v) (v)

18 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 8 Vst: sh cos ch Now from (v), (v) ad (v), w gt sh ( ) Ths complts th proof () Lt sh tah (v) ( ) cosch ( ) tah tah () sh tah From () ad (), w gt tah sh tah sh tah sch sh sh sh sh () ( ) Ths complts th proof π sh ta ta, 4 QNo: Show that () ( ) () tah ( cos) ( cosc) () sch ( s ) cot π ta 4 Sol: () To show: ( ) sh ta, to show: π ta sh ta 4 RHS π sh ta 4 ta π 4 π ta 4 π π ta ta 4 4 ta ta ta ta

19 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 9 Vst: cos s cos s cos s cos s cos s s cos cos cos s cos s cos cos s ta cos Ths complts th proof cos s cos s cos s s s s cos () To show: tah ( cos) ( cosc) s cos cos cos LHS ( ) tah cos tah cos cos cos cos cos cos ( ) ( cos ) s / cos cot s ( cosc ) cosc cosc ( cosc) R H S Hc tah ( cos) ( cosc) Ths complts th proof () To show: sch ( s ) cot to show: s sc h cot

20 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos 0 Vst: RHS sc h cot cot cot cot cot Ths complts th proof cot ta Q No : Fd tah f 5 sh 5 Sol: Gv that 5 sh ( ) ( ) cot s cos s cos s cos s s cos ± ± 4 or Lt s frst tak tah Now lt s cosdr scod val tah 8 4 As As Hom Assgmts

21 Compl Nmbrs: Hyprbolc fctos ad Ivrs hyprbolc fctos Vst: *** *** *** *** *** *** *** *** ***