1999 by CRC Press LLC

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1 Poulri A. D. The Melli Trform The Hdboo of Formul d Tble for Sigl Proceig. Ed. Alexder D. Poulri Boc Rto: CRC Pre LLC, by CRC Pre LLC

2 8 The Melli Trform 8. The Melli Trform 8. Propertie of Melli Trform 8.3 Exmple of Melli Trform 8.4 Specil Fuctio Frequecy Occurig i Melli Trform 8.5 Tble of Melli Trform Referece 8. The Melli Trform 8.. Defiitio Exmple M{ f( t); } = F( ) = f( t) t dt 8.. Reltio to Lplce Trform By lettig the trform become 8..3 Reltio to Fourier Trform By ettig which implie tht t t M{ e u( t)} = e t dt = Γ( ) x x t = e, dt = e dx, = σ+ M{ f( t)} F( ) f( e x ) e x = = dx = L { f( e x )} β i 8.. we obti x x βx F () = fe ( ) e e dx M{ f( t); } { f( e x x = σ+ β = F ) e ; β} 999 by CRC Pre LLC

3 where βx F{ f( x); β} = f( x) e dx = Fourier Trform 8..4 Iverio Formul c+ f() t = Ft () c d where c i withi the trip of lyticity < Re < b. 8. Propertie of Melli Trform 8.. Sclig Property 8.. Multiplictio by t M{ f ( t); } f ( t) t dt f ( x) x = = dx = F( ) 8..3 Riig the Idepedet Vrible to Rel Power ( + ) M{ t f( t); } = f( t) t dt = F( + ) M{ f ( t ); } f ( t ) t dt f ( x) x, x dx F = = = > 8..4 Ivere of Idepedet Vrible M{ t f( t ); } = F( ) 8..5 Multiplictio by l t M{l tf( t); } = d ( ) d F 8..6 Multiplictio by Power of l t d M{(l t) f( t); } = ( ) d F 999 by CRC Pre LLC

4 8..7 Derivtive M d d f(); t = ( )( ) F( ) ( )! Γ() ( ) ( )( + ) L( ) = = ( )! Γ( ) 8..8 Derivtive Multiplied by Idepedet Vrible M t d d ( ) f(); t ( )() F() ( ) F (),() ( ) ( ) = = Γ + = + L + Γ() Exmple M t 8..9 Covolutio d f() t t df () t + ; F ( ) dt dt = c+ M{ f( t) g( t); } = FzG ( ) ( zdz ) c 8.. Multiplictive Covolutio Propertie of the Multiplictive Covolutio t du M{ f g} = M f gu ( ) ; ( ) ( ) u u = F G M t du { FG ( ) ( )} = f gu ( ) u u. f g= g f commuttive. ( f g) h= f ( g h) ocitive 3. f δ( t ) = f uit elemet t f u du gu ( ) u t d f g t d f g f t d = g dt dt = ( ) dt (l t)( f g) = [(l t) f] g+ f [(l t) g] δ( t) f = f( t) 999 by CRC Pre LLC

5 δ( t p) δ( t p ) = δ( t pp ), p, p > d δ( t) d f = ( t f) dt d 8.. Prevl Formul c+ f() t g() t = M{ f ;} M{; gd } c r+ f() t g () t t dt = M{}( f ) M {}( g ) d β β β where β + r M{ f}() β = f() t t dt 8.3 Exmple of Melli Trform 8.3. Exmple 8.3. Exmple + to M{ tut ( t)} = t + dt=, Re{ } < t + M. + = t + t t ; dt t dt Settig t + = we obti: x =, dx = Hece, x t + ( + t). x dx M{ f ; }= ( x) = x ( x) dx = B(, ) = Γ( ) Γ( ) = ( x ) ( x ) i < Re{ } < Exmple m Γ( m) Γ( ) From ( u) u du =, Re{ } >, Re{ m} >, with the ettig u = x/( + x), we obti o Γ( m+ ) Hece, x m+ ( + x) Γ( m) Γ( ) dx =. Γ( m+ ) Γ() Γ( ) M{( + t) ; } =, < Re{ } < Re{ }. Γ( ) Exmple Uig 8..3 d we obti 999 by CRC Pre LLC

6 ( / ) Γ Γ b b M{( + t ) ; } =, < Re{ } < Re{ b}. Γ( b) Exmple M{ δ( t t ); } = δ( t t ) t dt = t o (ee 5.3.) Exmple From 8.3. d t M{ tut ( t)} = + d hece, 8.4 Specil Fuctio Frequecy Occurrig i Melli Trform 8.4. Defiitio + df t M F t + ; ( ) ( ) ( ) t dt = = + = = M{ u( t t ) t ; } + M{ t δ( t t ); } + The gmm fuctio Γ() i defied o the complex hlf-ple Re() > by the itegrl t Γ( ) = e t dt 8.4. Alytic Cotiutio The lyticl cotiued gmm fuctio i holomorphic i the whole ple except t the poit =, = L,,,, where it h imple pole Reidue t the Pole ( ) Re = ( ( )) = Γ! Reltio to the Fctoril Γ( + ) =! Fuctiol Reltio Γ( + ) = Γ( ) Γ() Γ( ) = i( ) 999 by CRC Pre LLC

7 Γ = / Γ( ) = Γ( ) Γ( + / ) (Legedre duplictio formul) m = m/ ( m)/ Γ( m) = m ( ) Γ( + / m), m =, 3, L, (Gu-Legedre multiplictio formul) Γ( ) ~ / exp + + O ( ),, rg( ) < (Stirlig ymptotic formul) The Bet Fuctio Defiitio: x y Bxy (, ) t ( t) dt Γ( x) Γ( y) Reltio to the gmm fuctio: Bxy (, ) = Γ( x + y) The pi Fuctio (logrithmic derivtive of the gmm fuctio) d Defiitio: ψ( ) l Γ() d = γ + = + + Euler cott γ, lo clled C, i defied by γ Γ ()/ Γ() d h vlue γ Riem Zet Fuctio ζ(, zq) = ( ), Re( z ) > q, q,,, L z + = ζ( z), Re( z) > z The fuctio ζ(z) i lytic i the whole complex z-ple except i z = where it h imple pole with reidue equl to by CRC Pre LLC

8 8.5 Tble of Melli Trform 8.5. Tble of Melli Trform TABLE 8. Some Stdrd Melli Trform Pir Origil Fuctio Melli Trform f (), t t > M f f t t [ ; ] ( ) dt Strip of holomorphy Algebric Fuctio b ut ( t ), > b+ b + Re( ) <Re( b) (( ut) ut ()) t b b+ b + Re( ) >Re( b) ( + t) i( ) < Re( ) < ( + t), rg < < Re( ) < Γ() Γ( ) ( + t) Γ( ) < Re( ) < Re( ) ( t) cot( ) < Re( ) < ( t), > < Re( ) < b u( t)( t), Re( b) > Γ() Γ() b Γ( + b) Re( ) > ut ( )( t) ( t + ), Re( ) > b Γ( b) Γ( b) Γ( ) Re( ) < Re( b) < cc < Re( ) < ( t + ), rg < ( / )cc( / ) / < Re( ) < ν t < t < t > h ν ( t ) < t < t h > Re( ν) > ( + ν) Re( ) >Re( ν) h Γ( ν) Γ h Γ ν + h Re( ) > ( t )( t ) + i cc cc < Re( ) < ( ) Expoetil Fuctio pt e, p > p Γ( ) Re( ) > ( e t ) Γ() ζ() Re( ) > (() ζ = zet fuctio) t ( e + ), Re( ) > Γ( )( ) ζ() Re( ) > t ( e ), Re( ) > Γ() ζ() Re( ) > ( e t )( e ), Re( ) > Γ( ) ζ(, ) Re( ) > ( e t ) Γ()[( ζ ) ζ()] Re( ) > 999 by CRC Pre LLC

9 TABLE 8. Some Stdrd Melli Trform Pir (cotiued) Origil Fuctio Melli Trform f(), t t > M[ f; ] f( t) t dt Strip of holomorphy t e h, Re( ) >, h > h / h Γ( / h) Re( ) > t e t e t e t Γ() e Logrithmic Fuctio Trigoometric Fuctio Γ( ) < Re( ) < Γ( / ) < Re( ) < t e h, Re( ) >, h > h / h Γ( / h) h < Re( ) < l( + t) ( / ) i( ) < Re( ) < < Re( ) < l( + t), rg < cc( ) < Re( ) < up ( )l( pt) p [ ψ( + ) + p l γ] Re( ) > t l( + t) l + t t lt < t < t > t ν l t < t < < t < e t (l t) ut ( )i( l t) u( t)i( l t) ( ut ( ) ut ( p))l( p/ t), p> ( )i( ) < Re( ) < ( / )t( / ) < Re( ) < (l ) Re( ) > ( + ν) Re( ) >Re( ν) d Γ( ) d + + Re( ) > Re( ) <Im( ) Re( ) > Im( ) p Re( ) > i( t), > Γ( )i( / ) < Re( ) < t / β e i( pt), Re( ) > Imβ ( + β ) Γ( )i t Re( ) > i ( t), > Γ( )co( / ) < Re( ) < co( t), > Γ( )co( / ) < Re( ) < t ( t) co t ( t) co( / ) co( / ) < Re( ) < < Re( ) < 999 by CRC Pre LLC

10 TABLE 8. Some Stdrd Melli Trform Pir (cotiued) Origil Fuctio Melli Trform f(), t t > M[ f; ] f( t) t dt Other Fuctio ν Γ + J ( ν t ), > ν Γ + i t Jν ( t), > δ( t p), p > p v ν Γ Γ v ν Γ( ) Γ Strip of holomorphy Re( ν) < Re( ) < 3/ < Re( ν) < Re( ) < / whole ple δ( t p), p > = p J () ν t Γ( + ν)/ Γ[( / )( ν ) + ] ζ( ) Re( ) < ν < Re( ) < 3/ r p δ( t p ),, l p δβ = r+ β l p = = p >, r = rel β=im( ) t b δ( b+ ) oe (lytic fuctiol) Referece Bertrd, Jcquelie, Pierre Bertrd, d Je-Philippe Ovrlez, The Melli Trform, i Trform d Applictio Hdboo, ed. Alexder Poulri, CRC Pre, Boc Rto, Florid, 996. Dvie, G., Itegrl Trform d Their Applictio, d ed., Spriger-Verlg, New Yor, NY, 984. Erdelyi, A., W. Mgu, F. Oberhettiger, d F. G. Tricomi, Tble of Itegrl Trfer, McGrw-Hill Boo Co., New Yor, NY, 954. Oberhettiger, F., Tble of Melli Trform, d ed., Spriger-Verlg, New Yor, NY, 974. Seddo, I N., The Ue of Itegrl Trform, McGrw-Hill Boo Co., New Yor, NY, by CRC Pre LLC

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