Re: Difficult integral: correction
- To: mathgroup at smc.vnet.net
- Subject: [mg23454] Re: Difficult integral: correction
- From: "Rasmus Debitsch" <debitsch at Zeiss.de>
- Date: Thu, 11 May 2000 00:54:06 -0400 (EDT)
- Organization: Customer of UUNET Deutschland GmbH
- References: <8f559t$49c@smc.vnet.net> <8fb0th$hid@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hello, with f =Expand[TrigToExp[Sin[x]^(2 rho)] Exp[I n x]] Integrate [f,{x,0,2 Pi}] i had no problem to compute the integral for rho=1,....,12 in a few minutes (Mathematica 4, WinNT, PIII 500, 256MB). Perhaps the TrigToExp is the trick. A other guess for the integral is : Let i[rho] = Integrate [(Sin[x])^(2 rho) Exp[I n x],{x,0,2 Pi}] and make the observation i[rho+1] = -2 i[rho] (rho+1) / (2rho+1)/(n^2-(2rho+2)^2) which will be a fast way to compute the integral for higher values of rho. Perhaps it is possible to prove this - I haven't tried it. Regards -- Rasmus Debitsch Carl Zeiss Lithos GmbH Carl Zeiss Strasse D-73447 Oberkochen eMail : debitsch at zeiss.de Marco de Innocentis <mdeinnocentis at hotmail.com> schrieb in im Newsbeitrag: 8fb0th$hid at smc.vnet.net... > I am sorry, I made a silly mistake: rho was supposed to be part > of the exponential, so > > > Integrate [(Sin[x])^[2 rho] Exp[I n x],{x,0,2 Pi}] > > Thanks, > > Marco > > In article <8f559t$49c at smc.vnet.net>, > Marco de Innocentis <mdi11 at hotmail.com> wrote: > > > > I need to calculate the following integral > > > > Integrate [(Sin[x])^2 rho Exp[I n x],{x,0,2 Pi}] > > > > as a function of the two parameters rho and n. I have tried it > > for rho = 1, 4, 8, 10, and it works, but it doesn't work at all > > for rho = 12. I left the computer on all night but only got the > > same expression as a result. > > Any suggestions? > > Thanks > > > > Marco > > > > Sent via Deja.com http://www.deja.com/ > > Before you buy. > > > > > > -- > http://www.thehungersite.com > > > Sent via Deja.com http://www.deja.com/ > Before you buy. > > begin 666 integral.nb M*"HJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ M*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ#0H-"B @(" @(" @(" @(" @ M(" @(" @36%T:&5M871I8V$M0V]M<&%T:6)L92!.;W1E8F]O:PT*#0I4:&ES M(&YO=&5B;V]K(&-A;B!B92!U<V5D(&]N(&%N>2!C;VUP=71E<B!S>7-T96T@ M=VET:"!-871H96UA=&EC82 T+C L#0I-871H4F5A9&5R(#0N,"P@;W(@86YY M(&-O;7!A=&EB;&4 at 87!P;&EC871I;VXN(%1H92!D871A(&9O<B!T:&4@;F]T M96)O;VL at #0IS=&%R=',@=VET:"!T:&4@;&EN92!C;VYT86EN:6YG('-T87)S M(&%B;W9E+ at T*#0I4;R!G970 at =&AE(&YO=&5B;V]K(&EN=&\ at 82!-871H96UA M=&EC82UC;VUP871I8FQE(&%P<&QI8V%T:6]N+"!D;R!O;F4@;V8 at #0IT:&4@ M9F]L;&]W:6YG. at T*#0HJ(%-A=F4 at =&AE(&1A=&$@<W1A<G1I;F<@=VET:"!T M:&4@;&EN92!O9B!S=&%R<R!A8F]V92!I;G1O(&$ at 9FEL90T*("!W:71H(&$@ M;F%M92!E;F1I;F<@:6X at +FYB+"!T:&5N(&]P96X at =&AE(&9I;&4@:6YS:61E M('1H92!A<'!L:6-A=&EO;CL-"@T**B!#;W!Y('1H92!D871A('-T87)T:6YG M('=I=&@@=&AE(&QI;F4@;V8@<W1A<G,@86)O=F4 at =&\ at =&AE#0H@(&-L:7!B M;V%R9"P at =&AE;B!U<V4 at =&AE(%!A<W1E(&UE;G4 at 8V]M;6%N9"!I;G-I9&4@ M=&AE(&%P<&QI8V%T:6]N+ at T*#0I$871A(&9O<B!N;W1E8F]O:W,@8V]N=&%I M;G,@;VYL>2!P<FEN=&%B;&4 at -RUB:70 at 05-#24D@86YD(&-A;B!B90T*<V5N M="!D:7)E8W1L>2!I;B!E;6%I;"!O<B!T:')O=6=H(&9T<"!I;B!T97AT(&UO M9&4N("!.97=L:6YE<R!C86X at 8F4-"D-2+"!,1B!O<B!#4DQ&("A5;FEX+"!- M86-I;G1O<V@@;W(@35,M1$]3('-T>6QE*2X-"@T*3D]413H at 268@>6]U(&UO M9&EF>2!T:&4 at 9&%T82!F;W(@=&AI<R!N;W1E8F]O:R!N;W0@:6X at 82!-871H M96UA=&EC82T-"F-O;7!A=&EB;&4 at 87!P;&EC871I;VXL('EO=2!M=7-T(&1E M;&5T92!T:&4@;&EN92!B96QO=R!C;VYT86EN:6YG('1H92 -"G=O<F0 at 0V%C M:&5)1"P@;W1H97)W:7-E($UA=&AE;6%T:6-A+6-O;7!A=&EB;&4 at 87!P;&EC M871I;VYS(&UA>2!T<GD at =&\ at #0IU<V4@:6YV86QI9"!C86-H92!D871A+ at T* M#0I&;W(@;6]R92!I;F9O<FUA=&EO;B!O;B!N;W1E8F]O:W,@86YD($UA=&AE M;6%T:6-A+6-O;7!A=&EB;&4 at #0IA<'!L:6-A=&EO;G,L(&-O;G1A8W0 at 5V]L M9G)A;2!297-E87)C:#H-"B @=V5B.B!H='1P.B\O=W=W+G=O;&9R86TN8V]M M#0H@(&5M86EL.B!I;F9O0'=O;&9R86TN8V]M#0H@('!H;VYE.B K,2TR,3<M M,SDX+3 W,# @*%4N4RXI#0H-"DYO=&5B;V]K(')E861E<B!A<'!L:6-A=&EO M;G,@87)E(&%V86EL86)L92!F<F5E(&]F(&-H87)G92!F<F]M( T*5V]L9G)A M;2!297-E87)C:"X-"BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ M*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*BHJ*0T*#0HH M*D-A8VAE240Z(#(S,BHI#0H-"@T**"I.;W1E8F]O:T9I;&5,:6YE0G)E86M4 M97-T#0I.;W1E8F]O:T9I;&5,:6YE0G)E86M497-T*BD-"B at J3F]T96)O;VM/ M<'1I;VYS4&]S:71I;VY;(" @(" V,3$U-"P@(" @(" @,3 at P.%TJ*0T**"I. 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