Pulrik A. D. Diribui, Del Fuci The Hbk f Frmul Tble fr Sigl Prceig. E. Aleer D. Pulrik Bc R: CRC Pre LLC, 999
5 Diribui, Del Fuci 5. Te Fuci 5. Diribui 5.3 Oe-Dimeil Del Fuci 5.4 Emple 5.5 Tw-Dimeil Del Fuci Referece 5. Te Fuci 5.. A Te Fuci ϕ( i rel-vlue fuci f he rel iepee vrible h c be iffereie rbirr umber f ime, which i ieicl zer uie fiie iervl. Emple 5. 5.. Prperie f Te Fuci. If f( c be iffereie rbirril fe, ψ( f(ϕ( e fuci.. If f( i zer uie fiie iervl, ψ( f( τϕ( τ τ, < <, i e fuci. 3. A equece f e fuci, {ϕ (} <, cverge zer if ll ϕ re ieicll zer uie me iervl iepee f ech ϕ, well ll f i erivive, e uifrml zer. Emple 5. [ ] < ep /( ϕ(, e fuci 0 ϕ( ϕ ϕ(. 4. Te fuci belg e D, where D i lier vecr pce uch h if ϕ D ϕ D, he ϕ ϕ D ϕ D fr umber. 999 b CRC Pre LLC
5. Diribui 5.. Defiii A iribui (r geerlize fuci g( i prce f igig ur rbirr e fuci ϕ( umber N g [ϕ(]. A iribui i l fucil. Emple 5.3 implie h u( i iribui h ig umber ech ϕ( equl i re. 5.. Prperie. Lieri-hmgeei: u( τϕ ( ϕ( N [ ϕ(] f g([ ϕ ( ϕ (] g( ϕ ( g( ϕ (. Shifig: 3. Sclig: g ( ϕ( g ( ϕ( g ( ϕ ( g ( ϕ 4. Eve iribui: 5. O iribui: g ( ϕ( 0, ϕ( g ( ϕ( 0, ϕ( eve 6. Derivive: 7. h erivive: g( ϕ( ϕ( g ( g ( ϕ( ϕ( ( g ( 8. Pruc wih rir fuci: prvie h f( ϕ( belg he e f e fuci. 9. Cvlui: [ g ( f(] ϕ( g ([ f( ϕ(] g ( τ g( τ τ ϕ( g( τ g( τ ϕ( τ 999 b CRC Pre LLC
Defiii A equece f iribui {g (} i i cverge he iribui g( if fr ll ϕ belgig he e f e fuci. 0. Ever iribui i he limi, i he ee f iribui, f equece f ifiiel iffereible fuci.. If g ( g( r ( r( (r beig iribui, he umber, he g ( g(, g ( r ( g( r(, g ( g(. A iribui g( m be iffereie m ime eire. The erivive f iribui lw ei, i i iribui. 5.3 Oe-Dimeil Del Fuci 5.3. Defiii lim g ( ϕ( g( ϕ( δ( 0 0 δ( ϕ( ϕ( 0, ϕ( i eig fuci 5.3. Prperie TABLE 5. Prperie f Del Fuci Del Fuci Prperie δ( δ( δ δ( δ( δ δ( δ( δ( δ( ; δ( eve fuci δ( f( f( 0 δ( f( f( f( δ( f( 0 δ( 999 b CRC Pre LLC
TABLE 5. Prperie f Del Fuci (ciue f( δ( f( δ( δ( 0 Aδ( Aδ( A Del Fuci Prperie f( δ( cvlui f( τ δ( τ τ f( δ( δ( δ( τ δ( τ τ δ[ ( ] N N N N N N δ( T δ( T ( N δ( T δ( f( f ( 0 δ( f ( f( δ( f( 0 f( ( δ( f ( 0 δ( f( δ( f( 0 δ( δ( (! δ (, m m m δ( m! δ( ( m m m, > m!, m < 0 δ( δ( 0, fuci δ( f ( f( δ( f( ( k 0 δ( δ( δ( u ( k k k! f( 0 δ( k k k!( k! δ( δ( δ( (, i eve if i eve, if i. δ( (i δ( 999 b CRC Pre LLC
TABLE 5. Prperie f Del Fuci (ciue δ( u( u( δ( u( δ( g( δ( Del Fuci Prperie δ( r( δ[(] r zer f r(, 0 r( δ( δ[(] r r( r( r zer f (, 0, 0 r( r( δ(i δ( π δ( δ( δ( δ( [ δ( δ( ] / ε e δ( lim ε επ ω δ( lim i ω π δ( lim ε π ε ε δ( ε lim ε 0 π ( ε δ( cωω π f ( [ u ( ( u ( u ( ] δ( u ( ( δ( u ( δ( cmb ( δ( T, f ( cmb ( f ( T δ( T T T COMB ω cmb ω δ ω ω π ω ( F { T ( } ( T jω lim e ω πδ( 999 b CRC Pre LLC
TABLE 5. Prperie f Del Fuci (ciue Del Fuci Prperie lim ( j j ( e ω ω ω π δ ( lim ( j j e ω δ ω ω π The fllwig emple will elucie me f he el prperie he ue f he el fuci. 5.4 Emple Emple 5.4 Equivlece f eprei ivlvig he el fuci: (c i δ( δ( b c i δ( c c e δ( e δ( Emple 5.5 The vlue f he fllwig iegrl re: ( 4 5 δ( 0 4 0 5 5, ( c δ( e δ( k k [ ( ( ] k k Emple 5.6 The fir erivive f he fuci i: 6 ( u ( u( ( u ( u[ ( ] δ( δ( ([ u ( ]c ( c u( c i δ( c u( i ( u( i δ( u π u u u π π ( π i δ δ( π i ( π c π u u π δ [ ( π]c 999 b CRC Pre LLC
Emple 5.7 The vlue f he fllwig iegrl re: e δ( ( [ e i ] i 4 4 4 6 ( ( 3 δ δ 3 δ( ( 3 ( 3 δ( 3 ( 3 ( ( 3 ( ( 6 5 4 9 Emple 5.8 The vlue f he fllwig iegrl re: 4 4 4 4 4 4 4 6 e 3 e 3 3 3 δ( δ e δ e e 4 4 4 4 4 6 e δ( 3 e δ[ ( 3] e δ( 3 e 5.5 Tw-Dimeil Del Fuci 5.5. Defiii δ(, δ( δ( δ(, δ( δ( 4 f(, ξηδ ( ξδ ( η ξη f(, A δ( δ( b p (, b, b he bur f A 5.5. Lie Me, A pa(, 0 herwie The fuci ϕ(δ( c be ierpree lie m he lie f ei ϕ(. Emple 5.9 p (δ( i lie m he -i wih ei e he -i frm. 999 b CRC Pre LLC
Emple 5.0 f(, δ( which i he prfile f f(,. 5.5.3 Lie M Curve α(, δ[α(,] i lie m he curve α(, 0 wih ei λ(, where α, α α α(, α. 5.5.4 Lie Me Alg - -Ae m α(, The lie me hve eiie lg he - -ireci give b α α m α(, m m α repecivel., α α α m m hece δ[α(,] δ( α(, 0 i he curve f α(,. α α 5.5.5 Slui f α(, If we lve α(, 0 fr ee i h r wih i he we m regr δ[α(,] he lie m imilrl fr he lui Emple 5. If δ[ r ] he α(, r, α /, α /,, ± r,, ± r f(, ξη δ( ξ ξη f(, η η δα α δ α (, [ (, ] ( i, α i δα α δ α (, [ (, ] ( i, α. i δα [ (, ] δ( r δ( r r r δ( r δ( r, r. Al r δα [ (, ] δ( r δ( r r [ ] [ ] < [ ] Sice α he δ[α(,] δ(r r i rig el fuci wih ui ei α r lg r r.. 999 b CRC Pre LLC
Emple 5. b If δ(α b c, he α(, α b c, α, α b, hece c, b c b b δ(α b c c b b c δ δ b. 5.5.6 Trfrmi f Crie fr δ( b c (ee Figure 5. cθ i θ, iθ c θ, b θ, k b b, c θ, i θ, ( b k k k /, ( b / k. b b b δ( b c δ c δ( k c δ( where c/ k. k k k FIGURE 5. Emple 5.3 f(,δ( b c k f b b, δ( k k where k b m b b The ei lg hi lie i f k,. k k c/ k. 5.5.7 The Fuci δ( b c, b c : Frm (5.5.5 b c b c δ( b c, b c δ δ b c b δ c b b c δ b c b b b bc δ b bc c b δ b c b δ( D, 999 b CRC Pre LLC
5.5.8 The fuci f(,δ( b c, b c f(,δ( b c, b c f(, δ(,. See (5.5.7 fr he vlue f D,,. D 5.5.9 cmb( b c, b c cmb( b c, b c δ( b c δ( b c m m See (5.5.7 fr he vlue f D,,. 5.5.0 cmb( b, b 5.5. f(, cmb( b c, b c b bm m δ δ D D D D D m b bm m cmb( b, b D D D D δ δ D m b bm f cmb b c b c f D D D m (, (,, D D m b bm m δ δ D D D D 5.5. δ[α (,] δ[α (,] δα [ (, ] δα [ (, ] i δ( δ( α α i α α i where i, i re he crie f he iereci f he curve α (, α (,, Emple 5.4 (See Figure 5. α α(, α(, α (, α (,, α, α, α. δ[α (,] δ[α (,] δ( δ(. Ierec (, (,. α (,, α (,, α / /, α / /, α /, α / 0. 999 b CRC Pre LLC
Hece frm (5.5. δα [ (, ] δα [ (, ] [ δ( δ( δ( δ( ] δα [ (, ] δα [ (, ] δ( δ( δ( δ( fr <. FIGURE 5. Referece Gelf, I. M., e l., Geerlize Fuci, Vl. -6, Acemic Pre, New Yrk, NY 964-69. Hki, R. F., Geerlize Fuci, Chicheer, Egl, 979. Lighhill, M. J., Iruci Furier Ali Geerlize Fuci, Cmbrige Uiveri Pre, New Yrk, NY, 959. Pulrik, A. D., Sigl em, i The Trfrm Applici Hbk, Eie b A. D. Pulrik, CRC Pre Ic., Bc R, Flri, 996. 999 b CRC Pre LLC