UNIT I Integr Trnform. INTEGRAL TRANSFORMS Integr trnform re ued in the oution of prti differenti eqution. The choice of prticur trnform to be ued for the oution of differenti eqution depend upon the nture of the boundry condition of the eqution nd the fciity with which the trnform F() cn be converted to give the origin function f(x).. DEFINITION The integr trnform F() of function f(x) i defined I [ ] f( x) F () fxxdx ()(,) Where (, x) i nown function of nd x, ced the erne of the trnform : i ced the prmeter of the trnform nd f (x) i ced the invere trnform of F(). Some of the we nown trnform re given beow : (i) Lpce trnform. (, x) x e L{ f( x)} F () fxe () dx (ii) Fourier compex trnform. (, x) e ix b x F{ f( x)} F () fxe ( ) ix dx (iii) Hne trnform. (, x) x J n ( x) H n { f( x)} F () fxx () J n (x) dx where J n (x) i the Bee function of the firt ind of order n.
A TEXTBOOK OF ENGINEERING MATHEMATICS III (iv) Meine trnform. (, x) x (v) Hibert trnform. K(, x) M{ f( x)} Fx ( ) fxx ( ) dx F() x. FOURIER INTEGRAL THEOREM Thi tte tht f( x) x dx f( x ) ()co ( ) f t utxdtdu The integr on the right hnd ide i ced Fourier integr of f( x). Proof: We now tht Fourier erie of function f(x) in ( c, c) i given by ( ) f x + nx nx nco + bnin...() c c n where, n nd b n re given by n c c nt ftdt (), n ft ()co dt c c cc c c nt b n f()in t dt c c c Subtituting the vue of, n nd b n in (), we get c c c c c c n n nt nx nt nx f( x) ftdt () f()co t co dt f()in t in dt c + c + c c c c c c c nt nx nt nx () + ()co co + in in ftdt c f t c dt c c c c c c n c n ftdt () f()co t ( t xdt ) c + c c c n c c c n f() t + co ( tx) dt n c Since coine function re even function i.e., co( θ) co θ, the expreion...() + n n co ( t x) co ( t x) c c n n
INTEGRAL TRANSFORMS Therefore, () become ( ) () co ( ) c n f x f t c t x dt c c () co ( ) c n f t t x dt c c c...() Now ume tht c incree indefinitey, o tht we my write n u nd du. c c Thi umption give n im co ( t x) co ut ( xdu ) c } c c Subtituting in () from (), we obtin co ut ( xdu )...() { } f( x) f() t co ut ( xdu ) dt...(5) Thu, f( x) f()co t ut ( xdudt ). Proved. Note: The foowing condition on f( x). (i) f (x) i defined inge-vued except t finite point in ( c, c). (ii) f (x) i periodic outide ( c, c) with period c. (iii) f (x) nd f'(x) re ectiony continuou in ( c, c). (iv) f() x dxconverge, i.e. f (x) i boutey integrbe in (, ).. FOURIER S COMPLEX INTEGRAL Proof: iux iut f( x) e du fte () dt We now tht fxdx ( ) f( x ) i odd function. in ut ( xdu ) [Since, in u (t x) i odd]
A TEXTBOOK OF ENGINEERING MATHEMATICS III Obviouy we hve ftdt () in ut ( xdu ) or i ftdt () in ut ( xdu ) (Mutipying by i) On dding () in R.H.S. of Fourier Integr Theorem, we hve...() i f( x) f()co t ut ( xdudt ) + ftdt () in ut ( xdu ) ftdt () [co ut ( x) + iin ut ( x)] du iut ( x) () ftdt e du or iux iut f( x) e du fte () dt Retion () i ced Fourier Compex Integr....().5 FOURIER TRANSFORM We hve iux iut f( x) e du fte () dt ix e d f() t e dt (u )...() it Putting f() t e dt F () in (), we get...() In () F() i ced Fourier trnform of f( x). it ix f( x) e df.()...() In () f( x ) i ced Invere Fourier trnform of F()..6 FOURIER SINE AND COSINE INTEGRALS f( x) in uxdu ft ()inutdt (Fourier ine integr) f( x) co uxdu f()co t utdt (Fourier coine integr)
INTEGRAL TRANSFORMS 5 or Proof: We now tht co u (t x) co (ut ux) co u (t x) co ut co ux + in ut in ux Then eqution (5) of rtice., cn be written f( x) f()(co t utcoux + inutin ux) dudt Ce. When f() t i odd. f( x) + ft ()coutco uxdudt f()in t utinuxdudt...() f()co t ut i odd hence f()co t utcouxdudt Q For odd function fxdx ( ) nd for even function From (), we hve f( x) in uxdu ft ()inutdt The retion () i ced Fourier ine integr. Ce. When f() t i even. f()in t ut i odd. f()co t ut i even. From (), we hve fxdx ( ) fxdx ()...() f()in t utinuxdudt f( x) co uxdu f()co t utdt...() The retion () i nown Fourier coine integr..7 FOURIER SINE AND COSINE TRANSFORMS We now tht Fourer ine integr f( x) in xd ft ()intdt (u ) f( x) in xdf.()...() where F () ft ()intdt...()
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III where In () F() i ced Fourier ine trnform of f( x). In () f( x) i ced Invere Fourier ine trnform of F(). We now tht Fourier coine integr f( x) co xd f()co t tdt (u ) f( x) co xdf.()...() F () f()co t tdt...() In eqution () F() i ced Fourier coine trnform of f( x ). In eqution () f( x) i ced Invere Fourier coine trnform of F(). Exmpe : Expre the function Fourier integr. Hence, evute So. The Fourier Integr for f( x) i when x f( x) when x > f( x ) inλcoλx dλ. (U.P.T.U. ) λ f()co t λ( t xdtd ) λ co λ( t xdtd ) λ, < t< Q f() t,otherwie in λ( t x) λ dλ in λ( x) + in λ ( + x) d λ λ λ (By in C + in D formu) Thu inλcoλx f( x) λ d. An. λ or inλcoλx d λ f ( x ) λ
INTEGRAL TRANSFORMS 7 or inλcoλx λ, for x d λ, for x > < For x i.e., x ± which i point of dicontinuity of f( x), vue of integr +. An. Exmpe : Uing Fourier ine integr, how tht λ < x< λ, when x > co, when in( xλ ) dλ, < x < So. Let f( x), x > Uing Fourier ine integr, we hve f( x) in λx ft ()inλtdtdλ inλx inλtdtdλ coλt inλx dλ λ coλ in( xλ) dλ λ coλ, < x< in( x ) d f( x) λ λ λ, x> Proved. Exmpe : Find the Fourier ine integr for Hence, how tht f( x) e βx ( β> ) βx λinλx e d λ. (U.P.T.U. ) β +λ So. The Fourier ine trnform of f( x ) i f( x) in λxdλ ft ()inλtdt...()
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III Putting the vue of f( x) in (), we get βx βt e inλxdλ e inλtdt t e β inλxdλ βinλtλcoλt β +λ ( ) ( ) in xd λ λ λ + β +λ x e dλ or β +λ βx λinλ βx λinλx e dλ. An. β +λ Exmpe : Uing Fourier coine integr repreenttion of n pproprite function, how tht x cowx e dw. (x >, > ) + w So. We now tht Fourier integr i f( x) co uxdu f()co t utdt Putting the vue of f() t nd repcing u by w, we get t t e cowxdw e cowtdt t e + w cowxdw { cowt+ winwt} cowxdw cowxdw + + w + w or x cowx e dw. Proved. + w x Exmpe 5: Find the Fourier ine trnform of e. Hence, evute xinmx. dx + x (U.P.T.U. ) So. In the interv (, ), x i wy poitive, therefore, e x x e.
INTEGRAL TRANSFORMS 9 The Fourier ine integr of f( x) i { x } in F e e xdx F + () x (y) Now the invere ine trnform of F() i e x. Uing inverion formu for the ine trnform, we get x e F ()inxd Repcing x by m, we get e + m + inxd inmd xinmx dx + x xinmx m Hence, we get dx e + x. An. Exmpe 6: Find the Fourier trnform of x, if x f( x), if x > So. x < x<, f( x), x > The Fourier trnform of function f( x) i given by Subtituting the vue of f( x) in (), we get ix F () f( x). e. dx...() F () ( x ) e. dx Integrting by prt, we get ( uv uv uv + uv ) ix... F () x x + i i i ix ix ix e e e ( ) ( ) ( ) ( ) ( )
A TEXTBOOK OF ENGINEERING MATHEMATICS III e e e e + i i i i i i i i i i ( e + e ) + ( e e ) i ( co) + ( in i ) i [ co in] +. An. Exmpe 7: Find the Fourier trnform of f( x) nd hve evute () in( )co( x) dx;, x < ie.. < x < f( x), x > iex.. > or x< in (b) d. So. We hve the Fourier trnform of f( x) { } ix F f ( x) e fxdx ( )...() nd if, then ix ix ix fxe ( ) dx + fxe ( ) dx + e fxdx () ix ix ix e fxe ( ) dx e dx i in i i i i e e i ( ) in( ), [From ()] F{ f( x)} e.dx i.e. { } [ ] [ ] F f ( x) x ( )
INTEGRAL TRANSFORMS then () Now { } ix F f ( x) e fxdx ( ) F () ix f( x) Fe () d in( ), x e ix < d, x > But L.H.S. in( )co( x) i ininx d d in( )co( ) x d Since the integrnd in the other integr i n odd function of. in( )co( x), x < dx, x > (b) If x nd in (), then in d in in d or d. An. Exmpe 8: Find the Fourier compex trnform of f( x) if iwx e, < x< b f( x), x< x, > b So. We hve ix F{ f( x)} e fxdx ( ) ix b ix iwx ix e dx e e dx e. dx + + b b iwx ( ) iwx + ( ) e + e dx i( + w) b iw ( + ) ( + e e ) i + w iwb. An.
A TEXTBOOK OF ENGINEERING MATHEMATICS III Exmpe 9: Show tht the Fourier trnform of So. We now x e f( x) i e. x ix ix F{ f( x)} fxe ( ) dx e e dx e ( ) x + ix x ix e dx e dx ( xi) ( ) xi e dx e. e dx. y, e dy x i putting y, o tht dx dy e e. y e,ince dy Proved. Exmpe : Find Fourier coine trnform of the function So. We hve co x, < x < f()if x f( x), x > { } F f( x) f( x)coxdx c { ( )} F f x c f( x)co xdx+ f( x)coxdx co x.coxdx+.coxdx x xdx x xdx co.co [ co( + ) + co( ) ] in( + x ) in( x ) + ( + ) ( )
INTEGRAL TRANSFORMS in( + ) in( ) +. ( + ) ( ) An. Exmpe : Find the Fourier ine nd coine trnform of f( x) e x. So. The Fourier ine trnform of f( x) i F() f( x)in( x) dx x x e in( x) dx { in( x) co( x) } e + nd Fourier coine trnform i x e + + in( ) co( ) +. { x x } F () e.co( x) dx c x An. e x { co( x) + in( x) } + + + +. An. Exmpe : Find Fourier ine trnform of. x So. inx F dx x x inθ dθ [Putting x θ o tht dx dθ] θ inθ dθ. An. θ Exmpe : Find the Fourier coine trnform of x x f( x) e + e
A TEXTBOOK OF ENGINEERING MATHEMATICS III So. The Fourier coine trnform of f( x) i given by F () f( x)coxdx Putting the vue of f( x ),we get x x ( ) F () e + e coxdx x x + e coxdx e coxdx Q 6 +. + + + 9 + + 9. An. e Exmpe : Find the Fourier ine trnform of. x x x e e.cobxdx [ binbx cobx] + b x So. The ine trnform of the function f( x) i given by { } F f( x) f( x)in( x) dx F()...() x e F().in( x) dx. x Differentiting both ide w.r. to, we get d d x e x [ ] ( ) F() xcox dx e co( x) dx x e x { co( x) + in( x) } + d [ () F ] d + Integrting w.r. to, we get F() d tn + c +...() But for, we hve F() [from ()] Putting thee vue in the eqution (), we get C F() tn. An.
INTEGRAL TRANSFORMS 5 Exmpe 5: Find the Fourier trnform of f( x) x, if < x < x f( x) +, if < x<, otherwie So. The Fourier trnform of f( x) i ix x ix x ix F{ f ( x) } F () fxe ( ) dx e dx e dx + + Chnging x by x in the firt integr, we get x x ix e dx+ e dx ix ix ix ( e + e ) x x dx co( x) dx x inx cox co( ) co( ). + [ ] An. x Exmpe 6: Find Fourier coine integr of the function e. Hence how tht coλx x ; d e λ x λ + So. We now tht Fourier coine integr f( x) λ λ λ co xd f()co t tdt x λ λ λ t e co xd e co tdt t e +λ coλxdλ ( coλ x+λinλx) co λxdλ. +λ
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III coλx d λ +λ If coλx x. d e λ +λ Proved.. An. Exmpe 7: Find the Fourier ine nd coine trnform of, < x < f( x), x > So. Fourier ine trnform Fourier coine trnform F () f()in x xdx inxdx cox co F () + inx F () f()co x xdx coxdx in. An. Exmpe 8: Find the Fourier coine trnform of So. Fourier coine trnform x, for < x< f( x) x, for < x<, for x > F () f()co x xdx / xco xdx + ( x)coxdx /
INTEGRAL TRANSFORMS 7 / x ( x) ( cox) inx cox inx + ( ) / in / co / co in / co / + + + co co / +. An. Exmpe 9: Obtin Fourier coine trnform of x, for < x< f( x) x, for< x <, forx < So. Fourier coine trnform { } F f( x) f()co x xdx c (U.P.T.U. ) xco xdx ( x)coxdx +.coxdx in x co x in co ( ) x x x ( ) x + in co co in co + + co co co (co ) co co + co ( co ). An.
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III Exmpe : Find the compex Fourier trnform of dirc-det function δ( t). So. F{( δt )} e δ( t dt ) it Lt + h h h it e dt h it e Lt h i Lt e h i + h ih e ih Note: Dirc-det function δ( t ) i defined e i ( t ) Lt I( ht ) δ, where h θ e ince Lt. An. θ θ Iht (, ) h for < t < + h for t< nd t> + h Exmpe : Find Fourier ine nd coine trnform of () n So. () ( ) c n F x in xx. dx n ( ) n Fc x co x. x dx ( ) + ( ) ( co + in ) n n n F x F x x i x x dx x n. ( b) x [U.P.T.U. (Comp.) ] ix n e x dx n t t dt e i i ( ) n t t n e t ( ) i dt
INTEGRAL TRANSFORMS 9 ( i) n ( i) n ( i) Γ n n n n Γn Equting re nd imginry prt, we get F F c n ( x ) n ( x ) n n n co + iin Γ n co + iin Γn n n Γn n co n Γn n in. An. n (b) n F c F co x in. x An. Exmpe : Find the ine trnform of e e x x + e e x x So. We hve x x e + e F() in( x) dx x x e + e x x ix ix e + e e e dx e e i x x ( + ix ) ( + ) e e i x x e e ix dx ( i) x ( + ) e e i x x e e i x dx.tn + i tn i i i Q x x e e dx x x tn e e
A TEXTBOOK OF ENGINEERING MATHEMATICS III + i i in in i + i i co co + i i i + i in co in co + i i co i co in+ in( i) [inin i] [co( i i) + co ] in( i) inh. An. [co( i i) + co ] [coh+ co ] Exmpe : Find the ine nd coine trnform of n x xe. So. Let n f( x) xe n x ().in( ) x F xe x dx...() x We hve e.in( x) dx ( inxcox) e x + + i i + i Differentiting both ide w.r. to, n time, we hve n n d d n n x ( ) xe inxdx n ( i) n( + i) id d Putting rcoθ, rinθ! + i n ( ) ( ) ( n ) ( ) ( n n + + i i ) i n ( n+ ) ( ) n! ir in( n+ ) θ ( n ) n! r n+ in( n+ ) θ
INTEGRAL TRANSFORMS n x xe inxdx n! ( ) ( ) in ( n tn ) n+ / + + Hence, from () r ( ) + nd θ tn Ao ( ) We hve n! F( ) in ( n+ tn ). An. n+ ( + ) F f( x)co( x) dx n x xe co( x) dx...() x x e e co( x) dx ( cox + xinx + + + i + i Differentiting both ide w.r. to, n time, we hve ( ) n n x n ( n+ co( ) ) ( )!( ) ( ) ( n xe x dx n i + + i + ) n x ( ) n+ n xe co( x) dx n! co( n ) + θ r Putting rcoθ, rinθ Exmpe : Show tht n! co ( n+ tn ) F () c ( ) ( n+ + ) /. An. () F [ xf( x) ] F ( ) (b) F [ xf( x) ] F ( ) d d c nd hence find Fourier coine nd ine trnform of xe. c x d d
A TEXTBOOK OF ENGINEERING MATHEMATICS III So. () c ( ) (b) ( ) F f( x)coxdx d F ( c ) xfx ( )in xdx d F { xf( x) } F f( x)inxdx d F ( c ) xf ( x )co xdx d F { xf( x) } x d x (c) Fc( xe ) F( e ) d c d d + (Uing exmpe ) ( + ) ( ) ( + ) ( + ) x x (d) F( e ) d Fc( e ) d dx d + (Uing exmpe ) ( + ). An. Exmpe 5: Find the invere Fourier trnform of y f( x) e, where y [ ],. (U.P.T.U. ) So. We hve, if, if { } nd ( ) F F f( x) Fe () d ix ix ix Fe () dx + Fe () d y ix y ix e e d + e e d
INTEGRAL TRANSFORMS e d e d ( yix) ( y+ ix) + ( yix) ( y ix) e + e + ( y ix) ( y+ ix) ( + ) ( ) { } { } y ix y ix y+ ix+ yix + y ix y + ix ( y ix)( y+ ix) y y + x y. An. y + x of Exmpe 6: Find the Fourier coine trnform of nd hence, find Fourier ine trnform + x x. (U.P.T.U. ) + x So. Fourier coine trnform F () f()co x xdx I c coxdx I...() + x di d xinx dx + x (+ x )inx dx x( + x ) inx dx + x inx dx x( + x ) di + d inx dx...() x( + x ) di d I cox dx ( + x )
A TEXTBOOK OF ENGINEERING MATHEMATICS III Soution of () i When di I d...() I ce + ce...() I + x di d From eqution () nd (5) c + c c c c, c Therefore, from eqution (), we hve di d ce ce...(5) dx [from ()] I e [from ()] cox dx ( + x ) e. An. Differentiting bove w.r.t., we get x xinx dx e ( + x ) inxdx e. An. ( + x ) Exmpe 7: Find the Fourier ine trnform of f( x). x ( + x ) So. We hve F ( ).in( ) x dx...() I (y) x ( + x )
INTEGRAL TRANSFORMS 5 Then di d d d.in ( ) x ( + x ) x dx di d co( x) dx...() ( + x ) di d xin( x) x in( x) dx dx ( + x ) x ( + x ) ( x + )in( x) dx x ( + x ) in( x) in( x) dx+ dx x x ( + x ) + I di ( ) I D I d d where D. d Then the oution of the bove differenti eqution i I Ae + Be +...() di Ae + Be...() d Now from () when, we hve I nd from () when So putting in () nd (), we get di dx x tn d + x I, A + B nd A + B di dx...(5)...(6)
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III Soving eqn. (5) nd (6), we get B, A Putting the vue of A nd B in eqn. (), we get x I ( e + e ). An. PROBLEM SET.. Expre, for x f( x), for x > Fourier ine integr nd hence evute. Uing Fourier integr, how tht coλ in λxdλ. λ winwx dw e ( + w ) x, x >. Show tht the Fourier trnform of, for x<α f( x) x, for α< x<β, forx >β i β x α i e e i. i. Show tht the Fourier trnform of x, for x < f( x), for x > > An. i ( co ). Hence how tht 5. Find Fourier trnform of f(x) if in t t dt. x, x f( x), x > An. i ( co in ) 6. Show tht the Fourier trnform of x e i ef reciproc.
INTEGRAL TRANSFORMS 7 7. Find Fourier ine nd coine trnform of x., An. x 8. Find Fourier trnform of e, >. An. + 9. Find Fourier ine nd coine trnform of co hx in hx. An., + +. Find the Fourier ine nd coine trnform of e αx βx + be. b α bβ An. +, + + +β +α +β.8 PROPERTIES OF FOURIER TRANSFORMS (i) Liner Property If F () nd F () re Fourier trnform of f (x) nd f (x) repectivey, then Ff [ ( x) + bf()] x F() + bf() where nd b re contnt. Proof: We now tht ix F() e. f( x) dx nd F() e. f() xdx (ii) Chnge of Sce Property Ff [ ( x) + bf( x)] e [ f( x) + bfdx( x)] dx ix ix ix ix e. f ( xdx ) + b e. f() xdx () + bf (). Proved. F If F() i the compex Fourier trnform of f( x), then { ( )} F f x F Proof: We now tht F () e. fxdx ( ) ix ix dt F{f(x)} e. fxdx ( ), put x t dx
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III (iii) Shifting Property e t i dt f() t i t e ftdt () F. Proved. If F() i the compex Fourier trnform of f( x ), then i F{ f( x )} e F () Proof: F () e. fxdx ( ) { } ix (iv) { } ix ix F f ( x e. f( x dx ), [put x t, o tht dx dt] it ( + ) e F e f( x) F ( + ) ix Proof: { ( )} i it ftdt () e e ftdt () i e F (). Proved. ix ix F e f x e fxe ( ) dx F ( + ). Proved. ix ( + ) e fxdx ( ) (v) Modution Theorem (U.P.T.U. 5) If F() i the compex Fourier trnform of f (x), then F{ f( x)co x} F ( + ) + F ( ) [ ] Proof: We now tht F () e. fxdx ( ) F{ f( x)co x } e. f( x)coxdx ix ix ix ix ix e + e e. f( x) dx. ( ). e ix fxe ix + e ix e ix dx
INTEGRAL TRANSFORMS 9 n (vi) If { } { } ( ) ix ( + ) ix ( ) e. fxdx ( ) e fxdx ( ) + F ( ) F ( ) + + n d F f ( x) F (),then F x f( x) i F (). n d Proof: We now tht [ F ( ) F ( ) ] + +. Proved. n ix F () e. fxdx ( )...() Differentiting () w.r.t. both ide, n time, we get n df () n ix ( ix) e. fxdx ( ) n d n n n ix () i ( x) e. fxdx ( ) n Fx ( f ()) x ( i) (vii) { } Proof: { } n () i F( x f()) x n d n { F ()} d F f ( x) if(), if f() x x ± ix F f ( x) e. f () xdx n ix ix ix { } { } e. d f () x dx e f( x) e.( i) fxdx ( ) i i e. fxdx ( ) if(). x F () (viii) F{ fxdx ( ) } ( i ) Proved. Proof: Let f( x) fxdx ( ) f ( x) f( x) x { ( )} ( ) () ( ) { ( )} F f x i F i F f x x if{ fxdx ( ) }
A TEXTBOOK OF ENGINEERING MATHEMATICS III x { ( ) } ( i) { ( )} F fxdx F f x F (). Proved. ( i) ( i) F{ f ( x) } Note: F () nd F c () re Fourier ine nd coine trnform of f(x) repectivey..9 CONVOLUTION The convoution of two function f( x ) nd g(x) i defined f( x) gx ( ) fugx ()( udu ) Convoution Theorem on Fourier Trnform The Fourier trnform of the convoution of f()nd x gx ( ) i the product of their Fourier trnform i.e., Proof: We now tht [ ( ) ( )] [ (). ] [ ()] F f x gx F f x Fgx f( x) gx ( ) fugx ()( udu )...() Ting Fourier trnform of both ide of (), we hve F[ f( x) gx ( )] F fugx (). ( udu ) ix fugx (). ( udu ) e dx fudu () { gx ( ue ) } ix du By inverion {(). f u dufg. ( x u)} iu fudue (). G () G (). fue () iu du G(). F() F(). G(). Proved. { (). ()} { ()} { ()} F FG f g F F F G (uing hifting property)
INTEGRAL TRANSFORMS of xe x Exmpe : Find Fourier coine trnform of. e x So. Firt of we find Fourier coine trnform of x { } Differentiting w.r.t., we hve nd hence evute Fourier ine trnform e x x Fc e e coxdx I...() di d xe x inxdx x ( xe ) inxdx x { in } x e xdx coecxe dx xe x coxdx Integrting, we hve di I d + di d I ogi + oga But when, from (), we hve When from (), we hve I A Hence, I Ae...() x I e dx A By chnge of ce property x { } x I e coxdx e. Fc e e An.
A TEXTBOOK OF ENGINEERING MATHEMATICS III We hve x e coxdx e Differentiting w.r.t., we get / xe x inxdx e x / xe in xdx e. { } x / F xe e An. Exmpe : Find the Fourier trnform of x e. Hence, find the Fourier trnform of (i) (iii) f( x) e x (U.P.T.U. ) (ii) ( x ) f( x) e (iv) f( x) e x / f( x) e cox x So. F () e. fxdx ( ) { } ix x ix x ( x ix). F e e e dx e dx i x + e i x dx e e dx, Put e e d e e d i x, dx d x { } F e (i) By chnge of ce property { } e...() x F e e e
INTEGRAL TRANSFORMS (ii) Putting in (i) prt, we get (iii) From eqution () x e F e x { } F e x F{ e } x { } e e. (By chnge of ce property) e 6 ( ) i 6 F e. e e (By hifting property) e i 6 x (iv) F{ e } e (By modution Theorem). PARSEVAL S IDENTITY FOR FOURIER TRANSFORMS (U.P.T.U. 5) If the Fourier trnform of f( x ) nd gx ( ) be F () nd G () repectivey, then (i) FGd () () fcgxdx ( ) ( ) where G () i the compex conjugte of G () nd gx ( ) i the compex conjugte of g(x). (ii) [ F ( )] d f() x dx Proof. (i) fxgxd ( ) ( ) f() x Ge () ix d d Since gx ( ) Ge () ix dx fxgx ( ) ( ) Gd (). fxe ( ) ix dx
A TEXTBOOK OF ENGINEERING MATHEMATICS III ix ince fxe ( ) dx F () Fourier Trnform Putting gx ( ) f( x) in (), we get GFd ()()...() FF (). () fxfxdx ( ) ( ) [ F ()] d [ ] f( x) dx. Proved.. PARSEVAL S IDENTITY FOR COSINE TRANSFORM (i) (ii) F c Gc d (). () fxgxdx ( ). () Fc () d f ( x ) dx. PARSEVAL S IDENTITY FOR SINE TRANSFORM (i) (ii) F G d (). () fxgxdx ( ). () F () d f ( x ) dx Exmpe : Uing Prev identity, how tht So. Let dx ( x + ) x f( x) e otht Fc() + By Prev identity for coine trnformtion Fc () d f ( x ) dx x x x e d e dx e dx ( + )
INTEGRAL TRANSFORMS 5 d. An. ( + ) Exmpe : Uing Prev identity, how tht xdx ( x + ) x So. Let f( x) x + o tht ( ) F() e By Prev identity for ine trnformtion F () d f ( x ) dx x x + dx e d e e d +. Proved. Exmpe 5: Uing Prev identity, prove tht dt b ( b) ( + t ) ( b + t ) x So. Let f( x) e, gx ( ) e Then bx + b Fc(), G () + b + c By Prev identity for Fourier coine trnformtion F (). () ( ). () c Gc d fxgxdx...() On ubtitution in (), we get b + b + d x bx. e e dx
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III b d ( ) ( b ) + + + b x ( ) e dx ( + b) x e + ( + b) + b d ( + ) ( b + ) b+ b dt. Proved. ( + t ) ( b + t ) b( + b) Exmpe 6: Uing Prev identity, prove tht int e dt t ( + t ) Proof: Let, < x < f( x), gx ( ) e x, x > F c in ( ), Gc( ) By Prev identity for coine trnform F c Gc d (). () fxgxdx ( ). () in x d f( x). e dx + ( + ) in + x x fxe ( ) dx+ fxe ( ) dx x. e dx + x e e e d ( e )
INTEGRAL TRANSFORMS 7 or int ( + t ) ( e ) dt. Proved. t Exmpe 7: Uing Prev identity, prove So. By exmpe, we now tht int dt. t If, for x < f( x), for x > > in Then F () Uing Prev identity f() t dt F () d ( ) in dt d in d Putting t, we get int dt t int A dt t dt Q d dt d int dt t. Proved. Exmpe 8: Sove for f(x) from the integr eqution. f( x)coxdx e So. f( x)coxdx e c { ( )} F f x e...()
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III f( x) F c e e e e + x coxd coxd { cox + inx}. x. An. +. FOURIER TRANSFORM OF DERIVATIVES We hve redy een tht n n { ( )} ( ) ( ) F f x i F u F i Fux ( ) u,where u x i Fourier trnform of u w.r.t. x. (i) ( ) { } (ii) F { f ( x) } f() + F ( ) c L.H.S. f ( x ).co xdx co xd { f ( x )} f( x)co x + f( x)inxdx { } (iii) { ( )} in [ ( )] F( ) f()uming f( x) x F f x xd f x f( x)in x f( x)coxdx { } fc( ) F f x xd f x (iv) c{ ( )} co [ ( )]
INTEGRAL TRANSFORMS 9 { } + f ( x)co x f ( x)inxdx f () + F { f ( x) } ( ) (v) { ( )} in [ ( )] F f () uming f( x), f ( x) x F f x xd f x c { } f ( x)in x f ( x)coxdx F { f ( x) } F [ () f() ] c F () + f () uming f( x), f ( x) x.. RELATIONSHIP BETWEEN FOURIER AND LAPLACE TRANSFORMS Conider t e gt (), for t > f() t, for t < Then the Fourier trnform of f() t i given by...() { } it F f () t e ftdt () e ftdt () it ( ixt ) pt e gtdt () e gtdt () where p x i L{g(t)} Fourier trnform of f(t)lpce trnform of g(t) defined by ()..5 FOURIER TRANSFORMS OF PARTIAL DERIVATIVE OF A FUNCTION u F Fu () where F(u) i Fourier trnform of u w.r.t. x. x u F u () x Fu () x F u u c F c () u x x x (ine trnform) (coine trnform)
A TEXTBOOK OF ENGINEERING MATHEMATICS III Proof: Let F[u(x, t)] be the Fourier trnform of the function u(x, t), i.e. ( ) The Fourier trnform of ix Fuxt, e uxtdx (,) u i given by x u ix u F e dx x x Integrting by prt, we hve u ix u ix u F e ie dx x x x ix u ix ix e ie u+ ( i) e udx x ix e. udx Agin integrtion u u, x when x Thu F Fuxt [ (,)] x u Simiry the Fourier ine trnform of u x i given by u F x u inxdx x or F uxt [ (,)] Fut [ (,)] x u x nd F F [ ut (,)] c u u x x x c (ine trnform) (coine trnform).6 APPLICATIONS OF FOURIER TRANSFORM OF HEAT CONDUCTION (TRANSFER EQUATIONS) In one dimenion het trnfer eqution, the prti differenti eqution cn eiy be trnformed into n ordinry differenti eqution by ppying fourier trnform. The required oution i then obtined by oving thi eqution nd inverting by men of the compex inverion formu. Thi i iutrted through the foowing exmpe.
INTEGRAL TRANSFORMS Exmpe 9: Sove the eqution u u t x ubject to the condition (i) u when x, t >, x >, t >, < x < (ii) u when t, x (iii) u(x, t) i bounded (U.P.T.U. ) (Note. If u t x i given, te Fourier ine trnform nd if u x t x i given, ue Fourier coine trnform.) So. In view of the initi condition, we ppy Fourier ine trnform u t inxdx d dt u inxdx x uin xdx u () + u() u when x du du uor + u dt dt u Ae t...() u ut (,) uxt (,)inxdx u u (,) ux (,)inxdx cox co u (,).inxdx...() From () putting the vue of u (,) in (), we get co A co t u e or co t u e d u u Exmpe : Sove t x initi condition u(x, ), x. for x, t under the given condition u ut x, t > with
A TEXTBOOK OF ENGINEERING MATHEMATICS III So. Ting Fourier ine trnform u u F F t x (, ) d u u + u t dt u+ u, where u i the fourier ine trnform of u. Thi i iner in u. du u u dt + t t u t ue u e dt e + c...() Since, u(x, ), u (, ). Uing thi in () By inverion theorem, u u + c c u ( ) ( ) u e e u e t t t u e t uxt (,) in xd. An. u u Exmpe : Sove for x<, t> given the condition t x u (i) ux (,) for x (ii) (, t) (contnt) x (iii) u(x,t) i bounded. So. In thi probem, the given eqution. u t x i given. Hence, te Fourier coine trnform on both ide of x u u Fc Fc t x du u u (, t) dt x
INTEGRAL TRANSFORMS u+ [Uing condition (ii)] du u dt + Thi i iner in u. Therefore, oving t t e t ue e dt c + ut (,) + ce t Since u(x, ) for x u (,) Uing thi in (), we get...() u (,) c + c Subtituting thi in () By inverion theorem ut (,) ( ) t e e uxt (,). co xd. An. t Exmpe : Ue Fourier ine trnform to ove the eqution under the condition u u t x (i) u(, t) (ii) u(x,) e x So. The given eqution i (iii) u(x, t) i bounded. u u t x...()
A TEXTBOOK OF ENGINEERING MATHEMATICS III Ting Fourier ine trnform on bothide of eqn. (), we get u in u in xdx t x xdx u u, where du () x dt u uinxdx du u dt +...() t It oution i u ce, when ci contnt. At t, ( ) ( ) u u in xdx t t From (), ( ) u x e + From () & (), c + inxdx...() c t...() t From (), u e + Ting invere Fourier ine trnform, we get t uxt (,) e in xd. + Exmpe : Ue Fourier coine trnform to how tht the tedy temperture n in the emiinfinite oid y > when the temperture on the urfce y i ept t unity over the trip x t ero outide the trip i tn + x tn x + y y r my be umed. x The reut e x inrxdx tn ( r, > ) < nd So. Ting Fourier coine trnform of u u +, we hve x y
INTEGRAL TRANSFORMS 5 u.co u xdx+.co xdx x y u d ( ) where x x dy u+ u u ucoxdx u du u dy...() Q x x It oution i But u i finite o c Otherwie u y y y +...() u ce ce From (), u ce y...() u ucoxdx y y ( ) ( ) From (), ( ) u u co xdx u y c From (), in.coxdx...() c in in u e y Appying invere Fourier coine trnform, we get y in y e u e coxd ( incox) d in( + ) + in( ) y e x x d + x x tn + tn y y
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III PROBLEM SET.. Appy pproprite Fourier trnform to ove the prti differenti eqution v v ; x >, t > t x Subject to condition v t (ii) v( x,) (i) (, ) x (iii) v(x, t) i bounded. x, x, x > in co t And. vxt (, ) + e cotd. Sove the eqution for high votge emi-infinite ine with the foowing initi nd boundry condition. v(x, t) nd i(x, ), v(, t) v u(t), v(x, t) i finite x. And. v vut [ x LC ], for x t LC nd v for t x > LC.7 FINITE FOURIER TRANSFORMS The finite Fourier ine trnform of f(x), <x< i defined ( ) ( ).in px F p f x dx ; p I Simiry, the finite Fourier coine trnform of f(x), < x < i defined ( ) ( ).co px Fc p f x dx ; p I Genery, the choice of the upper imit of integrtion in thee trnform i found convenient nd cn eiy be rrnged by hving uitbe ubtitution to ctu probem, then F( p) f().in x pxdx nd Fc ( p ) f ( x ).co pxdx
INTEGRAL TRANSFORMS 7.8 INVERSE FINITE FOURIER TRANSFORMS Inverion formu re given foow : When upper imit i For ine trnform: For coine trnform: px f( x) F( p) in p where Fc() tnd for fxdx ( ). When upper imit i. For ine trnform: px f( x) Fc() + Fc( p) co p For coine trnform: f( x) F( p) in px p where Fc() tnd for fxdx ( ). f( x) Fc() + Fc( p) copx p Exmpe : Find the finite Fourier ine nd coine trnform of (i) f( x ) in (, ) (ii) f( x) (iii) f( x) x in (, ) x in (, ) (iv) f( x ) in < x < / in / < x < (v) f( x) x in (, ) x (vi) f( x) e in (, ) So. (i) px copx cop F( p) F().in dx ; if p p p inpx Fc ( p).co pxdx ( ). p p An.
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III (ii) F ( p) F ( p) (iii) F ( p) F ( x) c px xin dx px px co in ( x) () (co p) p p p p ( ) ; if p. An. p c px xco dx px px in co p () ; if p p p p ( x) ( ) F ( p) F ( x ) ( x ) x px in dx px px px co in co ( x) () + p p p p ( ) + ; if p p p p ( ) Fc ( p ) px ( x ) co dx ( x ) px px px in co in ( x) () + p p p p ( ) ; if p p An.
INTEGRAL TRANSFORMS 9 (iv) { } F f( x) in pxdx + ( )inpxdx copx copx + p p p p co + copco p p p co p co + p ; if p { } F f( x) copxdx copxdx c inpx inpx p in ; if p p p p An. (v) px F( x ) x in dx ( ) px px px px co in co in ( ) + 6 (6) p p p p ( x ) x ( x) p p 6 p ( ) ( ) ; if p p + p px Fc x x co dx px px px px in co in co ( ) 6 (6) + p p p p ( x ) x ( x)
5 A TEXTBOOK OF ENGINEERING MATHEMATICS III p 6 p ( ) ( ) ; if p. p p An. (vi) x x px F( e ) e in dx x e px p px in co p + x e p p p + ( ) x Fc ( e ) p p + + x e px p px co in + p + e ( ) p ( ). An. p p + + Exmpe : Find finite Fourier ine trnform of f( x ) in (, ) So. The finite Fourier ine trnform of f(x) i given by x F( p) inpxdx x copx copx. dx p p inpx. An. p p p p
INTEGRAL TRANSFORMS 5 Exmpe : Find finite Fourier coine trnform of co ( x) f( x) in co ( x) F p co pxdx in So. c ( ) in co { ( ) + } + co { ( ) } x px x px dx in( x + px) in( x px) in p p+ p p+ p,,,,... An. Exmpe : Find finite Fourier ine nd coine trnform of f( x), x < x<. So. (i) px F p x dx ( ).in px px co co x. dx p p (ii) ( ) ( co p ), p p, p.co px Fc p x dx px px in in x. dx p p
5 A TEXTBOOK OF ENGINEERING MATHEMATICS III px co 8 inp p p p ( p ) p co, p When p, F ( p) F (). xdx 6. c c An. Exmpe 5: Find f(x) if it finite Fourier ine trnform i given by (i) cop F( p) p for p,,,. nd < x < (ii) p 6( ) F( p) for p,,,.nd < x < 8 p (iii) p co F ( ) p (p+ ) So. By inverion formu, for p,,, nd < x <. (i) cop f( x) inpx p p co p.inpx p p (ii) px f( x) F( p)in p p p 6( ) px in 8 p 8 p p ( ) px in p 8 (iii) px f( x) F( p)in p
INTEGRAL TRANSFORMS 5 p co in( px ); in. p ( p+ ) An. Exmpe 6: Find f( x) if it finite Fourier coine trnform i (i) p Fc( p) ; for p,, p ; for p given < x < (ii) p 6in cop F ( ) c p ; for p,,,. (p+ ) ; for p given < x < (iii) p co Fc( p) ; for p,,,.. (p+ ) ; for p given < x < So. By inverion formu, px f( x) Fc() + Fc( p).co p (i) Here F () /nd c p px f( x) + in co p p p p px + in co 8 p (ii) Here F c () nd p 6in cop px f( x) + co (p+ ) p
5 A TEXTBOOK OF ENGINEERING MATHEMATICS III (iii) Here F (), c p f( x) + co co (p+ ) p + co( px ) p ( px) p co. An. (p+ ) PROBLEM SET.. Find the finite Fourier ine trnform of (i) f( x) x in (, ) (ii) ƒ(x) co x in (, ) (iii) f( x) e x in (, ) p+ ( ), p An. () i p, p p p ( ii) [ ( ) co ] p. Find finite Fourier coine trnform of x e p p p ( iii) ( ) + p p + + x, < x <.. Find f(x) if it finite Fourier ine trnform i p for,,,, < x <. ( ) /, p An. Fc( p) / p, p p ( ) An. f( x) inpx p p
INTEGRAL TRANSFORMS 55.9 FINITE FOURIER SINE AND COSINE TRANSFORMS OF DERIVATIVES p F{ f ( x) } Fc( p) p p Fc{ f ( x) } ( ) f() f() F( p) p p p F{ f ( x) } F( ) () ( ) () p + f f p p Fc{ f ( x) } F ( ) ()( ) () c p + f f Proof: (i) F { f ( x) } f ()in x dx in. d{ f ( x) } px px px pxp f( x)in f( x).co dx (ii) c{ } p F c( p ) px px p px F f ( x) f ()co x dx f( x)co f( x). in dx (iii) { } p ( ) ( ) p f f() + F( p) px F f ( x) in df [ ( x)] ( ) in ( ) px p px f x f x co d p p ( ) n f ( ) f ( ) + f ( p) p ( ) + ( ) p p F p [ f() f()]
56 A TEXTBOOK OF ENGINEERING MATHEMATICS III (iv) c ( ) px F { f x } co d f ( x ) px p px f x f x in dx ( ) co + ( ) p p p ( ) f () f () + f ( p) c Note: If u u(x, t), then p p Fc( p) + f f () ( )( ) u p F F u x c( ) u p Fc F u u t + ut x Exmpe : Uing finite Fourier trnform, ove ( ) (, ) ( ) p (, ) u p p ( ) p F + ( ) ( ) ( ) F u u, t ut, x u p u u Fc F ( ) + (, ) co (, ) c u t p t x x x u u t x. Given u (, t) nd u (, t) nd u(x, ) x where < x <, t >. So. Since u (, t) given, te finite Fourier ine trnform. u in px dx u in px dx t x d u dt u F x p p n u + [(, u t) ( ) u(, t)] 6 A p uuing u(, t), u(, t) 6
INTEGRAL TRANSFORMS 57 du u p 6 dt Intergrting Since u(x, ) x p ogu t+ c 6 u p t 6 Ae...() px u( p,) ( x)in dx Uing () in (), Subtituting in (), By inverion formu, cop p u( p,) A co p p u p ( ) p p t 6 e p p+ 6 px e. An. p p uxt (, ) ( ) in...() Exmpe : Sove <, t >. v v ubject to the condition v(, t), v(, t), v(x, ) for < x t x So. Since v (, t) i given, ting F.F.S.T. of the given diff. eqution v inpxdx t v inpxdx x inpx dt x t dv v v copxdx p vcopx + p vin pxdx { }
58 A TEXTBOOK OF ENGINEERING MATHEMATICS III dv dt Integrting fctor ( ) p co p + pv p co p pv ( ) ( co ) + pv p p e pdt e ( co ) pt pt ve c+ p p e dt v c+ p( cop) e pt pt p ( p ) pt co ce +...() p Ting F.F.S.T. of the condition v(x, ) co px v( p,).inpxdx p Putting t in eqn. (), we get co p v( p,) c+ p cop cop c+ p p ( co ) cop p + p p p or cop cop c p p cop p Therefore, from eqution (), we hve cop pt cop v(,) pt e + p p Ting invere F.F.S.T., we get co p pt co p vxt (,) e + inpx p p p
INTEGRAL TRANSFORMS 59 pt p e inpx + inpx p cop co p p p cop pt cop + e inpx in px. p p p p An. Exmpe : Sove u u t x, < x < 6, t > u u Given (, t), (6, t) nd ux (,). x x x u So. Since (, t) i given, ue finite Fourier coine trnform t 6 6 u co px dx u co px dx t 6 x 6 d p u u u (6, t)co p u (, t) p + u dt 6 x x 6 c c c du u c c p 6 dt p oguc t+ c 6 uc u(x, ) x. At t Uing thi in (), we get Subtituting in (), we get p t 6 Ae...() 6 px 7...() 6 p u ( p,) ( x)co dx (cop) c 7 uc ( p,) A (cop) p 7 uc (,) pt (co p) p
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III By inverion formu, px uxt (,) fc ( ) + fc( p)co p 6 p t ( ) (co ) 6 xdx p e.co p p 7 px + 6 6 6 p p p t 6 (co p) px 6 + e.co. 6 An. PROBLEM SET. u u. Sove, < x <, t > given (, t) ; t x U(, t) ; u(x, ) in x in 5x. Ue the finite coine trnform to ove [An. t 5 t uxt (,) e inxe in5 x ] u u t x (one dimenion het eqution) u With the boundry condition when x nd x, t > nd the initi condition x U f (x), when t, < x <. p t An. uxt (,) fydy () + e co( px). f()co y pydy p u u for < x <. Sove, < x < 6. Given tht u(, t) u(6, t) nd u(x, ) t x for< x < 6 An. uxt p co px e p 6 p p t (,) 6 in
INTEGRAL TRANSFORMS 6. Z-TRANSFORMS Z-trnform i very uefu in the dicrete nyi. Difference eqution re formed in dicrete ytem nd their oution nd nyi re crried out by -trnform. Sequence Sequence {f()} i n ordered it of re or compex number. Repreenttion of Sequence Firt Method: The eementry wy i to it the member of the equence uch {f()} {5,, 7,,,,,, 6} The ymbo i ued to denote the term in ero poition i.e., i n index of poition of term in the equence. {g()}{5,, 7,,,,,, 6} Two equence {f()} nd {g()} hve me the term but thee equence re not identicy the me the eroth term of thoe equence re different. In ce the ymbo i not given, then eft hnd end term i conidered the term correponding to. Exmpe : {8, 6,,,,,, 5}, here The eroth term i 8, the eft hnd end term. Second Method: The econd wy of pecifying the equence i to define the gener term of the equence {f()} function of. f Exmpe : ( ) Thi equence repreent Exmpe :...,,,,,,... f( ), then {()} g {,,,,,,, } Bic Opertion on Sequence Let {f ()} nd {g()} be two equence hving me number of term. Addition: {f ()} + {g()} {f() + g()} Mutipiction: Let be cer, then {f ()}{ f ()} Linerity: {f ()}+b{g()}{ f () + b g()}
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III. DEFINITION OF Z-TRANSFORM { }. The Z-trnfrom of equence {f()} i denoted Z f( ) It i defined ( ) { } () ( ) Z f F f Where. i compex number.. Z i n opertor of Z-trnform.. F() i the Z trnform of {f ()}. Exmpe : If f () {5,, 7,,,,,, 6}, then f( ) 6 Z { f( ) } F () 5 + + 7 + + + 5 + + + Exmpe 5: Find the -trnform of { }, So. { } Z + + + +... + + + +... which i in G.P. whoe um r { }. An.. PROPERTIES OF Z-TRANSFORMS Linerity Theorem : If f () nd {g ()} re uch tht they cn be dded nd b re contnt, Then Proof: { } { ( ) + ( )} { ( )} + { ( } Z f bg Z f bz g Z f( ) + bg( ) f( ) + bg( ) [By definition]
INTEGRAL TRANSFORMS 6 f( ) + bg() f ( ) + b g () Z[{f()}+bZ[{g()}]. Proved.. CHANGE OF SCALE Z f F Theorem: If Z[{f ()}]F(), then { ( )} Proof: { } Subtituting for, we get But { } From () nd () F () Z f( ) f ( ) F f( )...() Z f( ) f( ) f( ) { ( )}...() Z f F. Proved.. SHIFTING PROPERTY Theorem: If Z [{f ()] F (), Then { } ± n Z f( ± n F () ( ) Proof: Z{ f( n} ( ) ± f n n f( n ) ± n.5 MULTIPLICATION BY K ± ± ± r ( n) ± n r fr () ± n F (). Proved. Theorem: If Z { f( ) } F (), then { } d Z f( ) F () d ±
6 A TEXTBOOK OF ENGINEERING MATHEMATICS III Proof: Z{ f( ) } f( ) f( ) f( ) ( ) In gener n { }.6 DIVISION BY K d ( ) d f( ) f ( ) d d d F (). Proved. d d Z f( ) F () d n Theorem: If { } Z f( ) F (), then f( ) Z Fd () Proof: f( ) f( ) Z ( ) ( ) f f d f ( ) d f () d Fd () f( ) Z Fd ().7 INITIAL VALUE Theorem: If Z [{f ()}] F (), Then f() im F (). Proof: { } Z f( ) f ( ) F () f() + f() + f() +... F () Ting the imit,, we get f() im F (). Proved.
INTEGRAL TRANSFORMS 65.8 FINAL VALUE Theorem: im f( ) im( ) F ( ). Proof: { } Z f( + ) f( ) f( + ) f( ) n n F() f() F () im f( + ) f( ) im( ) F ( ) f() + im im f( + ) f( ) n By chnging the order of imit, we get.9 PARTIAL SUM Theorem: If { } n im( ) F ( ) f() + im im[ f( + ) f( )] n Z f( ) F (), F () Then Z f( n) n Proof: Let {g()} be equene uch tht n im f() + { f( + ) f( )} n im[ f () f ()+ f ()+ f () f ()+. + f (n+) f (m)] im f( n+ ) im f( n) im f( ) n n g ( ) f( n) n We re required to find Z[{f()}, We now tht g ( ) g ( ) f( n) f( n) f( ) n { ( )} { ( ) } { ( )} Z g g Z f { } { } Z g ( ) Z g ( ) F () n
66 A TEXTBOOK OF ENGINEERING MATHEMATICS III G () G () F () F () f( ) G (). Proved. n. CONVOLUTION Let two equence be {f()} nd {g()} nd the convoution of {f()} nd {g()} be {h()} nd denoted where { } Proof: Z-trnform of () i { h ( )} { f( ) }*{ g ( )} h ( ) fng ( ) ( n)...() n gnf ( ) ( n) { g ( )}*{ f( ) } n Zh { ( )} ( fng ( ) ( n) ( fng ( ) ( n ) n n f n g n f n G () FG ( ) ( ) n n n ( ) ( ) ( ) ( ) n n Exmpe 6: Find the Z-trnform of { }. So. { } Z +... + + + + [ + + + +...] Thee re two G.P. nd um of G.P. r +, < nd < + ( ) + ( ) + ( )( ) ( )( ). An. ( )( )
INTEGRAL TRANSFORMS 67 Exmpe 7: Find the Z-trnform of So. Z Putting, we hve Z, < < ( )() 5 An. Exmpe 8: Find the Z-trnform of unit impue So. { }, δ ( ), Z f( ) δ( ) [ + + + + + + ]. An. Exmpe 9: Find the Z-trnform of dicrete unit tep, < U ( ), ( ) ( ) + + + +... So. Z { U } U Thi i G.P. it um i r. An.
68 A TEXTBOOK OF ENGINEERING MATHEMATICS III Exmpe : Find the Z-trnform of {ƒ()} where 5, < f( ), So. { } Z f( ) 5 + Thee re G.P. 9 7... + 5 + 5 + 5 + + + + +... 5 + + 5 5 + 5. An. (5 )( ) + 5 Exmpe : Find the Z-trnform of in α,. So. { in } iα α i e e Z α inα i Thee re G.P. iα α i iα iα e e ( e ) ( e ) i i i i iα iα α i iα ( e ) ( e ) ( e ) ( e )...... i + + + + + + i ie ie i e i e iα iα iα α i α i iα ( ) ( ) ( ) ( ) e e iα α i i e e i iα α i e e ( iα α i ) ( ) e e inα. An. i iα α i e + e + coα+
INTEGRAL TRANSFORMS 69 Exmpe : Find Z-trnform of in ( + 5). So. i(+ 5) i(+ 5) e e F () in( + 5) i i ( 5) + i ( + 5) e e i i 5 ( i i ) 5 i ( i e e e e ) i i 5i i i 5i i i e ( e ) ( e ) e ( e ) ( e )...... i + + + + + + i i5 e 5i e S i i i e i e r ( ) ( ) i i ( e )( e ) ( ) i 5 i 5 i i i 5 5 i i i e e e e e e e + e i i i i e e + i5 5i i i e e e e i i + + i i ( e e ) ( co) in5 in + in5 in co+, >. An. Exmpe : Find the Z-trnform of co +α 8 So. Z co +α co +α 8 8 co coαin inα 8 8 co coα in inα 8 8 coα co inα in 8 8
7 A TEXTBOOK OF ENGINEERING MATHEMATICS III co in coα 8 inα 8 co + co + 8 8 co co in in co co co in in 8 α α α α+ α 8 8 8 co + co + 8 8 coαco α 8. An. co + 8 Exmpe : Find the Z-trnform of coh +α So. F() +α +α e e + coh +α +α α α α e + e e e + e e α e α e e... e e e... + + + + + + + (Geometric erie um r ) α e α e + e e α α e + e e e e e
INTEGRAL TRANSFORMS 7 α α α α+ e + e e + e. cohαcoh α. coh + e e + cohαcoh α. An. coh + Exmpe 5: Find the Z-trnform of,. So. We now tht Z { } for the given equence, by the ce chnge formu the Z-trnform Z{. }. An. Exmpe 6: Find the Z-trnform of c in α,. So. We now tht Z { inα } inα coα+ By ppying the formu of chnge of ce we get (ee exmpe ) { inα } Z c inα c coα+ c c cinα ccoα+ c. An. Exmpe 7. Find the Z-trnform of (i) c coh( α), (ii) e co( α), (U.P.T.U. )
7 A TEXTBOOK OF ENGINEERING MATHEMATICS III So. (i) coh( ) α α e + e Z{ α } By chnge of ce property, α α ( e ) + ( e ) ( α e ) ( α + e ) c { coh( α ) } Z c + α e e α ( ) α α e + e ( coh ) α α α ( e + e ) + cohα+ cohα c cohα+ c c ( cohα) ccohα+ c. An. (ii) co( ) α α i i e + e Z{ α } iα α i ( e ) + ( e ) ( iα ) ( α i e + e ) + iα i e e α coα coα coα+ coα+ ( )
INTEGRAL TRANSFORMS 7 By chnge of ce property, { co( α ) } Z e coα e e coα+ e e ( ) ecoα ecoα+ e. An. Exmpe 8: Find the Z-trnform of { n C } ( n). n n n n n n n n... n Z C C + C + C + C + + C So. { } Thi i the expnion of Binomi theorem. ( + ) n. An. Exmpe 9: Find Z-trnform of { +n C n }. So. { n} + n + n n Z C + n> n C > + n C n n ( Cr Cn r) n+ n + n+ C C C + + + +... ( ) ( n+ )( n+ ) ( n+ )( n+ )( n+ ) + n+ + + ( ) +...!! ( )( ) ( n)( n ) ( ) ( ) ( ) ( n) n ( n) + n + + +...!! Thi i the expnion of Binomi theorem. n ( ) ( ) ( n + ). An. Exmpe : Find the Z-trnform of. ( )! So. Z!!
7 A TEXTBOOK OF ENGINEERING MATHEMATICS III Thi i exponenti erie. ( ) ( ) ( ) e + + + +...!!!! e. An. Exmpe : Find the Z-trnform of (i) f( ), (ii) f( ), ( + ) So. (i) { } Z f( ) f ( ) Z +... + + og ; if < og ; if > (ii) Z Z ( + ) + Z Z + og + og... + + + og + + +...
INTEGRAL TRANSFORMS 75 og og og og ( )og Exmpe : Find the Z-trnform of f *g where (i) f( n) un ( ), gn ( ) un ( ) n n n (ii) f( n) un ( ), gn ( ) un ( ) uing convoution theorem. So. (i) { } By convoution theorem n F () Zun ( ). ; if > n { } n n G () Z un ( ) ; if > { } { } n Z f* g Zh ( ) FG (). () n (ii) { } By convoution theorem { } { }. ; if > ( )( ) F () Z un () ; if > n { } G () Z un ( ) ; if > Z f* g Zh ( ) FG (). (). ; if >. ( )( ) An. An. PROBLEM SET.5 Find the Z-trnform of the foowing for ( ):.. An., > in. in. An. co+
76 A TEXTBOOK OF ENGINEERING MATHEMATICS III.. inh. in +α. An. inh coh + inα+ coα An., > + cinhα 5. c inh( α). An. ccohα+ 6. co ( α). 7. ( α) coh. ( coα) An. coα+ ( cohα) An. cohα+ 8. co +. An. ( + ) 9. coα + bin α. ( co in ) αb α An. coα+. inh(7 ). An. inh7 coh7+. co (9).. coh(5 ). ( coh9) An. coh9+ ( coh5) An. coh5+ 9. co + 5. co5co 5 An. co +
INTEGRAL TRANSFORMS 77. INVERSE Z-TRANSFORM Invere Z-trnform i proce for determining the equence which generte given Z-trnform. If f () i the Z-trnform of the equence {f ()}, then {f ()} i ced the invere Z-trnform of F(). The opertor for invere Z-trnform i Z. If Z{ f } ( ) F (),then { } Z f() f( ). METHOD OF FINDING INVERSE Z-TRANSFORMS We hve the foowing method of finding invere Z-trnform. Convoution Method.. Long Diviion Method.. Binomi expnion Method.. Prti Frction Method. 5. Reidue Method... Convoution Method We now tht { } Z f* g FG ( ) ( ) { } Z FG ( ) ( ) f * g fmg ( ) ( m) m Exmpe : Uing convoution theorem evute So. We now tht { } Z FG (). ( ) f * g Z. ( )( ) Let F () f( ) () nd G () Now, Z { FG (). ( )} ( ) * ( ) g ( ) ( ) () m m m m (which i G.P.)
78 A TEXTBOOK OF ENGINEERING MATHEMATICS III.. Long Diviion Method + +. An. ( ) ( ) ( ) + + Exmpe : Find Z. So. Ce I: > + + 8 8 Z 8 + + + +... + + + 8 +... + +... { } { }. An.
INTEGRAL TRANSFORMS 79 Ce II: < 8 6 + 8 8 8 6 6 Z... 8 6 + + + +... + +...... +... { + } { }. An... Binomi Expnion nd Prti Frction Exmpe : Find the invere Z-trnform of (i) > (ii) <.
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III So. Ce I: + + + +... Z + + + +... + +... { } { }. An. Ce II: < ( ) Z + + + +... +... { f( ) }. An. where, { f( ) }...,,,.. Prti Frction Method Here we pit the given F() into prti frction whoe invere trnform cn be written directy. Exmpe : Find the Z-trnform of (), (b) + 7+ ( ) ( ) So. () F () A B + + 7 + ( + )( + 5 ) ( + ) ( + 5) ( + ) ( + 5) F() ( + ) ( + 5)
INTEGRAL TRANSFORMS 8 { ( )} Z ( ) f Z F Z Z Z+ + 5 ( ) ( 5) (b) F() ( ) ( ) or F () ( ) ( ) Now F () ( ) ( ) D /, A 6, B nd C / A+ B+ C D ( ) + Z 6 + F() + ( ) + ( ) + + 8 F() ( ) ( ) + +. ( ) Now f( ) Z { F ()} { + } { } +. An. Q Z + ( ) Exmpe 5: Find the invere Z-trnform of. ( )( ) () <, (b) < < nd (c) > So. () < + + ( ) F() ( )...... + + + + + + + + +
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III (b)... + + + + +... < <, F() ( ) + ( )...... + + + + + + + +............ f( ) { } f { } (c) >..., > ( ),. An. + ( ) F() ( )...... + + + + + + + +...... + + + + { },, Exmpe 6: Find the invere of Z-trnform of ( 5), > 5. So. F() ( 5)
INTEGRAL TRANSFORMS 8 ( 5 ) 5 ( n ) + ( n+ ) n + 5 + 6(5 ) + (5 ) +... + (5 ) +... ( + )( + ) ( + )( + ) 5 5 Repcing by, we get F() ( + )( + ) ( )( ) 5 5 ( )( ) f( ) Z { F ()} 5,, < An. Exmpe 7: Obtin Z ( )( ), when () < < (b) <. So. () ( )( ) 6 6 6 ( ) ( ) 6 6 () ( ) 6 [+ + ( ) + ( ) +...]... + + + ( ) 6 ( ) ( ) ( ) 6 + 6 f( ) ; if > f( ). ; if < An.
8 A TEXTBOOK OF ENGINEERING MATHEMATICS III (b) < 6 6 6 6 ( ) ( ) ( ) ( ) 6 6 ( ) ( ) 6... 6... + + + + + + 6 + + ( ) +... 6 + + +... 6 6 6, + + Exmpe 8: Obtin Z +, ( ) ( ) when < <. So. Let + A B C + + ( ) ( ) ( ) ( ) ( ) Converting into prti frction, we get Z + + + ( ) ( ) ( ) ( ) ( ) + 9 ( ) ( ) ( ) ( ) 9 +......... 8 + + + + + + + + + + + + + 9 9 7 9 7 { } + + 8 + + + +...... + + + + +... 5, + F () if, nd, < or, + Z F () ( + ),. An.
INTEGRAL TRANSFORMS 85..5. Reidue Method Te the contour c uch tht the poe of the function ie within the contour. Then by reidue method f() um of the reidue of F () t it poe. where reidue for impe poe i ( ) F () Reidue of order n t the poe n d n ( ) () n i F ( n) d i Exmpe 9: Evute So. Z + 7+. F () + 7 + ( + )( + 5) Poe re given by, ( + ) ( + 5) or, 5 There re two impe poe. Conider contour > 5 Reidue (t ) ( ) ( + ) ( + )( + 5) Reidue (t 5) ( 5) ( + 5) ( + )( + 5) 5 f () um of the reidue ( ) ( 5) + { } Exmpe : Evute [ ] ( ) ( 5). An. Z. ( )( ) So. The poe re given by, ( )( ) nd There re two impe poe. Let u conider the contour >. Reidue t ( ) Reidue t ( ) ( ) ( )( ) ( ) ( )( ) Hence, f () Sum of the reidue. An.
86 A TEXTBOOK OF ENGINEERING MATHEMATICS III Exmpe : Evute Z 8 + 6. ( )( )( ) So. The poe re given by ( ) ( ) ( ),, There re three poe. Let u conider the contour >. ( ) ( 8+ 6) Reidue t ( ). ( )( )( ) 8 + 6 ]. 8. + 6. ( )( ) ( )( ) + +. 9. +. 6. +.. +. ( ) ( 8+ 6) Reidue t ( ) ( )( )( ) + 8 + 6 ] ( )( ) +. 8. + 6. ]. + 8. 6. ( ) 7. + 5. 6. +. Reidue t ( ) ( ) ( 8+ 6) ( )( )( ) + 8 + 6 ] ( )( ) +. 8. + 6. ]. 9. +.. 6. +. + Hence, f () Sum of the reidue + +, >. An. Exmpe : Evute Z 9. ( ) ( ) So. The poe re given by () ( ) /,. There re two poe i.e., one i impe poe t nd other i poe of order t /. Let u conider the contour >.
INTEGRAL TRANSFORMS 87 Reidue t ( ) ( ) 9 9 9. ( ) ( ) ( ) 5 + +. ( )..9 d d + 9 + ( + ) Reidue t ( /) d ().( ) d 9( ) ( ) + ( + ) ( + ) 5 5 9 Hence, f () um of the reidue. + ( + ). An. 5 5 Exmpe : Uing reidue method, evute Z 8. ( ) So. Here 8 F (). ( ) Poe re given by (poe of order ). Conider contour 5. Reidue of F() t ( ) d 8 ()! d ( ) ( ). { } d d d + ( 8) ( 8 ) d {( ) } ( ) d + + + + 8 ( ) 8( ) d ( )( )() 8( )() ( )( )() ( )() + + + + 7 ( ) ( 7 )() + + + + + + Z 8 ( ) { } um of the reidue ( ) + 7 + (). An.
88 A TEXTBOOK OF ENGINEERING MATHEMATICS III Exmpe : Uing reidue method, how tht + 5 Z (.) + (.). (5 )(5+ ) 75 75 So. Here, + F () (5 )(5 + ) Poe re given by (5 )(5 + ) i.e., /5. /5 which re impe poe. Conider contour + Reidue t ( /5) Lt. 5 (5 )(5 + ) 5 Lt ( + ) 5 (5 + ) 5 5 5 + (.) (5) 5 5 5 5 + 75 + Reidue t ( /5) Lt...( ) + Lt 5 5 + (5 )(5 + ) 5 5(5 ) Z ( ). 6.. +. + 5 5 5 (5) 6 {()} F ( ). (.) (.) 75 75 Hence, f () um of the reidue 5 (.) + (.). An. 75 75 Exmpe 5: Find Z. >. + So. The poe re determined by, + i.e., ± i There re two poe t i nd i. Let u conider contour >
INTEGRAL TRANSFORMS 89 Reidue t ( i) + + ( ) i. () i () i + + i i i i co + iin co iin + Reidue t ( i) + + ( + ) i. ( ) i ( i) + i i i i co iin co iin f () um of the reidue co + iin + co iin co co. An. PROBLEM SET.6. Find Z uing convoution theorem. ( )( b). Find the invere of the Z-trnform of the foowing: An. + + b b (), ` ( )( 5) < 5 An. 5 5 (b), ` ( )( ) < + + An. () ( )( ),( < ). Find the invere Z-trnform of (). >, (b) <. An. (){ }, () b { }. Evute Z 5, >. ( )( ) An. { } +,
9 A TEXTBOOK OF ENGINEERING MATHEMATICS III 5. Evute invere Z-trnform of the foowing: () e α (b) ( ), > An. An. { } α { e }, (c) + coα, > coα+ (d) og( ), > (e) e ( e ) > e An. { α} co, An., An. { e } 6. Find the invere Z-trnform of the foowing function by reidue method: () An., ( )( ) (b) (c) (d) + ( ) ( + ) ( 6+ ) ( ) ( ) An. { } An.co () i + ( i),,, + An. U (). DIFFERENCE EQUATION A difference eqution i retion between the difference of n unnown function t one or more gener vue of the rgument. nd Thu y( ) + y( )...() re difference eqution. n+ ( n+ ) y( n) n y +...() Eqution () my be written y y + y...() n+ n+ n yn+ yn+ yn+, yn yn yn y + + + n From eqution () yn+ yn + yn...()
INTEGRAL TRANSFORMS 9. ORDER OF A DIFFERENCE EQUATION The order of difference eqution i the difference between the rget nd the met rgument occurring in the difference eqution divided by the unit of increment. from eqution () Order Lrget rgument Smet rgument Unit of increment ( n+ ) n And order of eqution () i ( n+ ) ( n ).5 THEOREM y Z( y ) y y... n n + n n y, where Z( y ) y Proof: L.H.S. Putting + n m n ( + n) + n + n + n Z( y ) y y n Z( y+ n) ym m n m Zy ( ) y y Note: () For n,,,..., we hve Zy ( + ) Zy ( y) n n m m + n m m m m n y y y y y y Zy+ Z y y ( ) ( )... n n y y Zy ( + ) Z ( y y ) nd o on. () If Zy ( ) y, then Zy ( ) n n y.
9 A TEXTBOOK OF ENGINEERING MATHEMATICS III Exmpe : Sove by Z-trnform: y+ y+ + y ; y, y. So. Te Z-trnform on both ide Z( y ) Zy ( ) + Zy ( ) Z() + + y ( y y ) ( y y) + y ( ) + y y + ( )( ) Now, ting invere Z-trnform, we get y y Z { }, where,,. An. Exmpe : Sove by Z-trnform: 6y+ y y ; y, y. So. 6y+ y+ y Ting Z-trnform on both ide, we get Z[6 y y y ] + Z(6 y ) Z( y ) Z( y ) Z() + + 6( y y y ) ( y y ) y On putting the vue of y nd y, we get 6 y6 y y (6 ) y 6 6 6 5 5 6 6 y ( )( ) 6 + + +
INTEGRAL TRANSFORMS 9 6 6 5 5 6 6 y Z Z 5 5 + 6 5. An. Exmpe : Sove the difference eqution y+ y+ + y+ y U ( ), y y y By Z-trnform. So. y+ y+ + y+ y U ( ) Ting Z-trnform on both ide, we get Zy [ y + y y ] ZU( ) + + + Zy [ ] Z[ y ] + Zy [ ] Zy [ ] ZU( ) + + + [ y y y y ] [ y y y ] + [ y y ] y ZU( ) Putting the vue of y y y in the bove eqution y y + y y ( + ) y ( ) y y ( ) ( ) ( )( ) ( ) y coeff.of in coeff. of - in ( ) ( ) Exmpe : Ue Z-trnform to ove the difference eqution. n+ n+ n ( )( ),. An. 6 y y + y n+ 5 (U.P.T.U. ) So. yn+ yn+ + yn n+ 5...() Ting Z-Trnform in (), on both ide, we get Z{ y } Z{ y } + Z{ y } {} Zn + 5 Z{} n+ n+ n