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O  =e8(f MATHESIS Societ Italiana di Scienze Matematiche e Fisiche Sezione di Lanciano-Ortona 24 febbraio 2010 Ferdinando Casolaro - Universit del Sannio fcasolar@unisannio.it  I risultati di Eulero per le applicazioni della didattica di oggi Uf+fffE`F/ 0Strutture differenzialifUn insieme D in cui siano definite due operazioni binarie (+ e ) costituisce una struttura D(+, ), che detta struttura differenziale, se in D definita un operazione unaria tale che: 1) 2) 3) ffQff4ffLf,Strutture differenziali$fStrutture differenzialif 6Teorema di Liouville F. Casolaro  Decisione per integrali indefiniti  Atti del Convegno nazionale Mathesis  Cattolica 1991 (pag. 68-86) F. Casolaro   Il problema dell integrazione indefinita  Ratio Mathematica n. 4  1992 (pag. 29-38)pf$ "V(/ D6 > =Applicazioni del Teorema di Liouville alle funzioni razionali>>f  ? 2 TEOREMA DI LIOUVILLEf S 5Esempi di funzioni non integrabili per via elementare66f Z L integrale non esprimibile mediante funzioni elementari. "IGf([ Algoritmo parallelo di Risch:fa fL integrale non esprimibile mediante funzioni elementari.JJf lAlgoritmo parallelo di Risch:f}+Le funzioni goniometriche attraverso Eulero,,f% ~+Le funzioni goniometriche attraverso Eulero,,(%L integrale non esprimibile mediante funzioni elementari "USf( L integrale non esprimibile mediante funzioni elementariMMf   Analogie tra funzioni definite su figure omologiche Consideriamo la circonferenza , e l iperbole equilatera di equazione [F. Casolaro  F. Eugeni  Ratio Mathematica II (1996) pag. 23-33]; [F. Casolaro  L- Cirillo  Atti congresso naz. Mathesis  Verona 1996: pag. 309-318] lF 3ff[ft $ t Indicato con  la misura dell arco AP, il punto P ha coordinate: x = cos  y = sen  L area del settore circolare OAP data da:  = "r = , cio:  = 2  " 1a) Le coordinate di un punto P della circonferenza sono definite come funzioni del doppio dell area del settore circolare OAP individuato dal punto O origine degli assi, dal punto P considerato sulla circonferenza, e dal punto A origine degli archi.  ff7f$f4f ffffff?.  &(f$fL area del settore iperbolico OAQ data da:  = , cio:  = 2  " 1b) Le coordinate di un punto Q dell iperbole equilatera sono definite come funzioni del doppio dell area del settore iperbolico OAQ individuato dal punto O origine degli assi, dal punto Q considerato sull iperbole e dal punto A intersezione dell iperbole con l asse delle ascisse. b f3 fFfffFfFf f fFfFfFfFfFffv-&Trasformazioni che conservano la norma''fIl modulo | z | di un vettore , ovvero la norma, : Chiamiamo prodotto circolare (o interno) di due numeri complessi z1 e z2 di componenti rispettivamente (a1, b1) e (a2, b2) il numero: z1z2 = a1b1 + a2, b2 In tale modo l angolo di due vettori ovvero di due numeri complessi z1 e z2 ha un coseno dato da: @ ffqffff!ffff ffffffffffffffffHfffff y&Trasformazioni che conservano la norma''fI numeri complessi di norma unitaria sono costituiti dal cosiddetto  esponenziale complesso o  circolare definito ponendo Le due Trasformazioni: rappresentano, rispettivamente, i movimenti diretti e i movimenti inversi del piano. Infatti: cio: cio: ^CBf!ffffffffPPertanto, le due trasformazioni conservano la norma, in quanto, da: risulta: QQfImmediatamente si vede che, in tali trasformazioni, si conserva l angolo tra i due vettori. Infatti, indicati rispettivamente con gli angoli formati dai due vettori e dai loro complessi coniugati, da: risulta: f2L esponenziale iperbolicof Posto , tutte le relazioni ottenute sulle funzioni circolari, le ritroviamo sulle funzioni iperboliche. Infatti, la norma del numero iperbolico w = a+ib (con ) data da: che, con , rappresenta la norma del vettore unitario, in quanto verifica la relazione fondamentale delle funzioni iperboliche: I numeri iperbolici (o bireali) di norma unitaria sono costituiti dal cosiddetto  esponenziale bireale o  iperbolico definito ponendo :p(f>A#4Trasformazioni (iperboliche) che conservano la norma55fCome per il caso circolare, le due trasformazioni conservano la norma e conservano i settori iperbolici. Infatti, con passaggi analoghi a quelli fatti sulle funzioni circolari, si ha: da cui, si evince che:f $Prodotto interno e coseno iperbolico%%f In analogia al caso circolare, definiamo prodotto interno (iperbolico) di due numeri iperbolici W e W ,la grandezza: Da: risulta: cio: da cui si evince che W e W individuano due semirette caratterizzanti il settore iperbolico  di coseno iperbolico: Allora, la trasformazione conserva la norma iperbolica e il prodotto iperbolico.8{ZZf,z{   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> ` 4/(  4 4 6w " `}  `,Fare clic per modificare lo stile del titolo- -G 4 0z " `  uFare clic per modificare gli stili del testo dello schema Secondo livello Terzo livello Quarto livello Quinto livello: v 4 0 "^ `  Z*" 4 0܆ "^   \*" 4 0D "^ `  \*"H 4 0޽h ? 3380___PPT10.дܹ *Struttura predefinitao 0 (-(  ( ( 0 P    P*   ( 03     R*  d ( c $ ?  9 ( 0h2  0  uFare clic per modificare gli stili del testo dello schema Secondo livello Terzo livello Quarto livello Quinto livello: v ( 64= _P   P*   ( 6@ _   R*  H ( 0޽h ? 3380___PPT10.< ao 0N0 ,(  , ,  6@4S" I w   , s *4S" <  H , 0޽h ? 33___PPT10u..O̳+D=' = @B + o 0N0 0 1 1(   x  c $x84^j     c $>4q  "P@08X   0 ZA ? ?4"`'  d   6A ? ?"`-x d %  S A #??"?P x #d $  S A %??"?tL d x %d ) N ?"6@ NNN?N m ~v0___PPT106___PPT9 i ; 8 ""     / 0 c $0e0eAj      ? A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'p4A)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab      1  TA ? ?"6@ NNN?N t H  0޽h ? 33___PPT10i.Ugx+D=' = @B +Y o 0N0 X P 0  (  x"  c $TR4^     NS ?"6@ NNN?N"7B~v0___PPT106___PPT9 ? ""   NhY ?"6@ NNN?N"B~v0___PPT106___PPT9 @, ""    N$T ?"6@ NNN?N` ~v0___PPT106___PPT9 ? ""     H_ ?"6@ NNN?N_<___PPT10Z___PPT9<4 L insieme costituito da tutti gli elementi tali che viene chiamato Campo delle costanti di D. L insieme D delle funzioni razionali una struttura differenziale. wF  8f &f f) & f g f & f &  f & f g f & f & f " " 0 \A ?4?"?}  ?  d( 0  bA ? 4?"?]-] %  d( 0  bA ? 4?"?  d'H  0޽h ? 3fffff3̙3f̙___PPT10i.m-`+D=' = @B +r o 0N0 q i @  (  x"  c $e4     N  ?"6@ NNN?Nz~v0___PPT106___PPT9 ZL insieme:*  &f$f  S A 2??"?= bx 2d  H ?"6@ NNN?N~v0___PPT106___PPT9 Ncon R(x) funzione razionale, una struttura differenziale, in quanto risulta:nO&f&f&f&f&f$f  B ?"6@ NNN?N  S A  3??"?_ x 3d   NȖ ?"6@ NNN?N  ~v0___PPT106___PPT9 ZL insieme *  &f$f   S Ae T??"?c =s { x Td    H ?"6@ NNN?NF L ~v0___PPT106___PPT9 Gcon S(x) funzione razionale un campo differenziale in quanto risulta:nH&f&f&f&f&f$f   B ?"6@ NNN?N   S A  V??"?f WO.x VdH  0޽h ? 3fffff3̙3f̙___PPT10i.n`m+D=' = @B +% o 0N0 $$p"$(  x"  c $4      S A! Z??"?,x Zd   S AL \??"?$c<x \d   S A M??"?R` *x Md  S A N??"?b* x NdK  S A ??"?X Wj x d  S A ??"?A  Y x d  S A ??"?A _ x d  S A R??"? m x Rda   N ?"6@ NNN?Nd#2~v0___PPT106___PPT9 JSia *&f$f   H< ?"6@ NNN?N,4~v0___PPT106___PPT9 k%una funzione razionale degli elementi*&%&f$f  Nd ?"6@ NNN?N$ 6D~v0___PPT106___PPT9 v; se esiste una funzione >"&f"   N ?"6@ NNN?N$D~v0___PPT106___PPT9 ] tale che: 6 " &f$f  N ?"6@ NNN?N~v0___PPT106___PPT9 t", allora esistono degli elementi: 6#" &f$f  N9 ?"6@ NNN?N( = H ~v0___PPT106___PPT9 @, ""   N9 ?"6@ NNN?N( lH ~v0___PPT106___PPT9 D,& ""   N9 ?"6@ NNN?N 7 ~v0___PPT106___PPT9 ae delle costanti complesse *&f$f  N 9 ?"6@ NNN?N S ~v0___PPT106___PPT9 \ tali che:6 " &f$f  N9 ?"6@ NNN?N( ~ QH ~v0___PPT106___PPT9 F,& , ""   B ?"6@ NNN?N  S A ??"? -E x d  S A$ _??"?Y JIx _d  S A  ^??"?+ mCx ^d$  S A `??"?Y &Ix `d  H9 ?"6@ NNN?N 6~v0___PPT106___PPT9 Hcon (&f$f  H 9 ?"6@ NNN?N+ * K~v0___PPT106___PPT9 Zfunzione razionale di (&f$fz   H$9 ?"6@ NNN?N+ pK~v0___PPT106___PPT9 >,  "  ! N)9 ?"6@ NNN?N+ >K~v0___PPT106___PPT9 Yfunzioni irriducibili(&f$f " N/9 ?"6@ NNN?NK  k~v0___PPT106___PPT9 Xsul campo complesso.(&f$fH  0޽h ? 3fffff3̙3f̙___PPT10i.qkΙ+D=' = @B + o 0N0 | (  x"  c $4 `}   2  Hi9 ?"6@ NNN?N@ F |FLa derivata di una funzione razionale ancora una funzione razionale:GGf  B ?"6@ NNN?N  S A  ??"? x d   B ?"6@ NNN?Nww  S A% ??"? t x dAe   Nn9 ?"6@ NNN?N   JNon vale il viceversa.fH  0޽h ? 3fffff3̙3f̙___PPT10i.vP#+D=' = @B +M o 0N0 LD(  x"  c $t947^<  9   N8v9 ?"6@ NNN?NZ ~v0___PPT106___PPT9 PSia 0&f"   S A& ??"?/z @ !x dA  Hy9 ?"6@ NNN?NO7,; ~v0___PPT106___PPT9 una funzione razionale e sia p il grado di q(x). Se si conoscono le p radici (reali o complesse) di q(x), l integrale:x&f&f&f&f &f&f&f&f&f&f&f$f   B ?"6@ NNN?N   B ?"6@ NNN?N   S A( ??"?S % = x d#   NP9 ?"6@ NNN?Nc 4  g3 esprimibile mediante funzioni elementari e si ha:44f  B ?"6@ NNN?N   S A) ??"? -7 x d  S A* ??"?8 w(x d  S A+ ??"?8 @ (x d  S A, ??"?vdDfx dAh  H9 ?"6@ NNN?N 7+~v0___PPT106___PPT9 Hcon (&f$f  Hؗ9 ?"6@ NNN?N m +~v0___PPT106___PPT9 `funzione razionale; 0&f"   Nh9 ?"6@ NNN?N ++~v0___PPT106___PPT9 polinomio irriducibile su C; N&fgf&f$fH  0޽h ? 3fffff3̙3f̙___PPT10i.v "+D=' 9= @B + o 0N0 $(#(  x"  c $94^  9   B ?"6@ NNN?N  B ?"6@ NNN?N  S A- ??"?, x dA   NԪ9 ?"6@ NNN?N  Mcon n>2*   B ?"6@ NNN?Ntt   S A. ??"?OdnGx d   B ?"6@ NNN?N   S A/ ??"?x d  B ?"6@ NNN?Ntt   S A0 ??"?odvg x df  B ?"6@ NNN?N  S A1 ??"? e x d  B ?"6@ NNN?N  S A2 ??"?P  x d  B ?"6@ NNN?Nww  S A3 ??"?"P x d  B ?"6@ NNN?N  B ?"6@ NNN?N  S A4 ??"? d2 x d  B ?"6@ NNN?N  S A' ??"? ) x d  B ?"6@ NNN?N  S A5 ??"? P 8 x dd  B ?"6@ NNN?N   B ?"6@ NNN?N  S A7 ??"? xx d " B ?"6@ NNN?N ! S A8 ??"? dxx d $ B ?"6@ NNN?N # S A9 ??"? P x dA  & B ?"6@ NNN?N % S A: ??"? %x dA  ( B ?"6@ NNN?N ' S A; ??"?  x dH  0޽h ? 3fffff3̙3f̙___PPT10i.x+D=' 9= @B +j o 0N0 ia (  x"  c $(94 ^  9  0 BA ?4    9*  H9 ?"6@ NNN?N<  tDim. Se l integrale si potesse esprimere mediante funzioni elementari, dovrebbe esistere una funzione g(x), tale che: Pvfbfft  B ?"6@ NNN?Nttj  H ?"6@ NNN?N   >Per il 2 Teorema di Liouville, g(x) dovrebbe essere del tipo:8? fff !   B ?"6@ NNN?N``  S A< ??"?F Z> x dA   <D ?"6@ NNN?N   O(1)" lK2P   H ?"6@ NNN?N  Rin cui compare un solo addendo logaritmico in quanto il denominatore presenta l unico fattore irriducibile x.:nkfffk( 0  bA? ? 4?"?w] <  d 0 s *0e0eA     ? @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'p4A)BCD|E||s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab s   H  0޽h ? 3fffff3̙3f̙___PPT10i.y+D=' 9= @B +) o 0N0 ( (  x"  c $L4 `}     HT  ?"6@ NNN?N/7wO _)Derivando ambo i membri della (1), si ha:**f  B ?"6@ NNN?Ntt  S A= ??"?moex d  Nx& ?"6@ NNN?N  9cio:f   B ?"6@ NNN?N  S A> ??"?3 Sbx d   N* ?"6@ NNN?Nd  @ La funzione  f   B ?"6@ NNN?N   S A# c??"?8  x cd  B ?"6@ NNN?NK\K  H/ ?"6@ NNN?N  ~v0___PPT106___PPT9 D, di cui ipotizziamo l esistenza, 6#" &f$f  N3 ?"6@ NNN?N  ~v0___PPT106___PPT9 Tper cui si pu*&f$fH  <; ?"6@ NNN?NS s  4deve essere una combinazione lineare finita di x ed @5/ffP  <Q ?"6@ NNN?N V  ^esprimere nel seguente modo:fP  B ?"6@ NNN?N  S AA ??"?v 7 x d" 0 \A@ ?4?"?@ G   dH  0޽h ? 3fffff3̙3f̙___PPT10i.z X+D=' 9= @B + o 0N0 (    NXX ?"6@ NNN?N/Z O~v0___PPT106___PPT9 Vda cui, derivando:(&f$f  S AB ??"?gbx d4 0 # nAA ?G4?"?+8'K   dl  N_ ?"6@ NNN?N}  |Si sostituisce tale sviluppo nell equazione differenziale (2):??f   B ?"6@ NNN?N  S AC ??"?*T x dA( 0  bA> ?4?"?w @ ?  d  Ne ?"6@ NNN?Nu S(2)& lKfP  Hn ?"6@ NNN?N i  !Tale identit non pu mai essere verificata perch il termine della sola compare al primo membro con coefficiente 1, ma non nel secondo. Contraddizione che dimostra la non esistenza della funzione verificante la (1). f  B ?"6@ NNN?N  S AD ??"? k x d|  B ?"6@ NNN?N  B ?"6@ NNN?N  B ?"6@ NNN?N  S AF ??"? #  x dH  0޽h ? 3fffff3̙3f̙___PPT10i.{:Q+D=' 9= @B +  o 0N0     (  x"  c $`4 ^     B ?"6@ NNN?N  S AE ??"?Z0 @x d  B ?"6@ NNN?N  H4 ?"6@ NNN?N'v fDim. Se l integrale si potesse esprimere mediante funzioni elementari dovrebbe esistere una funzione g(x) tale che: Hufaff fq   B ?"6@ NNN?N   S AG ??"?s ; x dl   NP ?"6@ NNN?N v  <Per il teorema di Liouville, g(x) dovrebbe essere tale che: 6=fff "   B ?"6@ NNN?N   S AH ??"? ` x d  <씑 ?"6@ NNN?N S  O(1)" lK2PH  0޽h ? 3fffff3̙3f̙___PPT10i.} +D=' 9= @B + o 0N0 (  x"  c $4 `     B ?"6@ NNN?N  S AI ??"?9\x d  B ?"6@ NNN?N  S AJ ??"?X/x d   B ?"6@ NNN?N  S AK ??"?oj x dY   H ?"6@ NNN?NdB2H@8___PPT9 S ! 8XU   H ?"6@ NNN?NM7mD<4___PPT9 S ! 8XY   H Ñ ?"6@ NNN?N27RH@8___PPT9 S ! 8X@  N\Α ?"6@ NNN?N2tSR >Derivando la (1): si ha:??f    Nّ ?"6@ NNN?N   I(2)f    ND ?"6@ NNN?N d |0___PPT10<___PPT9 q Si sostituisce .!&f$f8X  S A  ??"? jL x dA  N ?"6@ NNN?N @  ~v0___PPT106___PPT9 ` nella (2):( &f$f    B ?"6@ NNN?N  S AM ??"?  x d  S AN ??"?( =r x d  S AO ??"? x d  B ?"6@ NNN?N&8&  N  ?"6@ NNN?Nt  " 0 \AH d?4?"?z m\ d dH  0޽h ? 3fffff3̙3f̙___PPT10i.+D=' 9= @B +S  o 0N0 R J   (  x"  c $4 ^      S AP ??"?tZx d  S AQ ??"?P `x d  S AR ??"?GQ x d  S AS ??"?~  x dA  S AT ??"?5 to x d  S AU ??"?c ~ s x d   N\ ?"6@ NNN?N    B ?"6@ NNN?N    S AV ??"? G] x d  S AW ??"?9 ~ vkx d  B ?"6@ NNN?NccH  0޽h ? 3fffff3̙3f̙___PPT10i.c+D=' 9= @B +e' o 0N0 d&\&  %(  x"  c $4 `}    0 c $0e0eA     ? A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'p4A)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab70    0 c $0e0eA     ? A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'p4A)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab70 j   0 c $0e0eA     ? A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'p4A)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab v    S AZ ??"? v x d  S A[ ??"?V :<x d  B ?"6@ NNN?N^^   B ?"6@ NNN?N '  0 c $0e0eA     ? A@  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'p4A)BCD|E||s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab F  H  0޽h ? 3fffff3̙3f̙___PPT10i.!+D=' 9= @B +h o 0N0 g_0(  r"  S  "4w     B ?"6@ NNN?N  S A\ ??"?w tx d  B ?"6@ NNN?Ntt   B ?"6@ NNN?NttM   N@% ?"6@ NNN?N  Dim. Dalle relazione di Eulero:* ff$   B ?"6@ NNN?Ntt   S A_ ??"?m ex d   N- ?"6@ NNN?NC 8si ha:  B ?"6@ NNN?Ntt  S A` ??"?dL x d  Nh2 ?"6@ NNN?N& 7 ~v0___PPT106___PPT9 Ponendo: 2ix=t, z  &f&f& f&f&f$f  S Aa ??"?  x d  Nx; ?"6@ NNN?Nb  ~v0___PPT106___PPT9 \ , risulta:6 "&f$f  B ?"6@ NNN?N  B ?"6@ NNN?Ntt  S A] ??"? m  x d  N`D ?"6@ NNN?Ne 7~v0___PPT106___PPT9 Kdove *&f$f  S Ab ??"? W%x dA  N+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D ' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* ++0+ 0 ++0+ 0 ++0+ 0 +| o 0  #(  x  c $\u4 `}     0  bAh ? 4?"? @    0 c $0e0eAy     ? 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'2 S5fcasolar@unisannio.it.--S5Sv-@Arial???-.  2 d.-@Arial???-. 2 dI risultati di .-@Arial???-. 2 d<Eulero.-@Arial???-. .2 dQper le applicazioni della .-@Arial5??-. !2 l<didattica di oggio.-@Arial5??-.  2 lm.-՜.+,D՜.+,0     QPresentazione su schermo g' #ArialBlackadder ITCTimes New RomanVerdana WingdingsStruttura predefinitaMicrosoft Equation 3.0Grafico di Microsoft GraphDiapositiva 1Strutture differenzialiStrutture differenzialiStrutture differenzialiTeorema di Liouville F. Casolaro Decisione per integrali indefiniti Atti del Convegno nazionale Mathesis Cattolica 1991 (pag. 68-86) F. Casolaro Il problema dellintegrazione indefinita Ratio Mathematica n. 4 1992 (pag. 29-38)>Applicazioni del Teorema di Liouville alle funzioni razionali2 TEOREMA DI LIOUVILLE6Esempi di funzioni non integrabili per via elementareILintegrale non esprimibile mediante funzioni elementari. Algoritmo parallelo di Risch:Diapositiva 11JLintegrale non esprimibile mediante funzioni elementari.Algoritmo parallelo di Risch:,Le funzioni goniometriche attraverso Eulero,Le funzioni goniometriche attraverso EuleroULintegrale non esprimibile mediante funzioni elementari MLintegrale non esprimibile mediante funzioni elementariDiapositiva 18F Analogie tra funzioni definite su figure omologiche Consideriamo la circonferenza , e liperbole equilatera di equazione [F. Casolaro F. Eugeni Ratio Mathematica II (1996) pag. 23-33]; [F. Casolaro L- Cirillo Atti congresso naz. 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