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Îäíàêî ïðèîïåðàòîðíîé òðàêòîâêå ïðåîáðàçîâàíèå Ôóðüå ðàñïðîñòðàíÿåòñÿ íà âåñüìàøèðîêèå êëàññû îáîáùåííûõ ôóíêöèé.Ïåðâûì êëàññè÷åñêèì ðåçóëüòàòîì î ïðåîáðàçîâàíèè Ôóðüå ÿâëÿåòñÿëåììà Ðèìàíà - Ëåáåãà.Òåîðåìà 2. Åñëè f (t) èíòåãðèðóåìàÿ ïî Ëåáåãó ôóíêöèÿ, òî åå ïðåîáðàçîâàíèå Ôóðüå fb(x) ðàâíîìåðíî íåïðåðûâíî íà âåùåñòâåííîé îñè R èfb(x) → 0 ïðè |x| → ∞. òåîðèè ïðåîáðàçîâàíèÿ Ôóðüå âàæíóþ ðîëü èãðàþò ðåçóëüòàòû îáîáðàùåíèè. Ïðèâåäåì îäèí èç íàèáîëåå ïðîñòûõ ðåçóëüòàòîâ.7.1. Îñíîâíûå òåîðåìû130Òåîðåìà 3.
Ïóñòü f (t) ∈ L1 (R), íåïðåðûâíà â òî÷êå t0 è óäîâëåòâîðÿåòâ íåé óñëîâèþ Äèíè:Zδ|f (t0 + h) − f (t0 )|dh < ∞.|h|−δÒîãäà çíà÷åíèå f (t0 ) âîññòàíàâëèâàåòñÿ ïî ôîðìóëå îáðàùåíèÿ1f (t0 ) = lim √R→∞2πZRfb(x)eit0 x dx.−RÎáû÷íî ôîðìóëó îáðàùåíèÿ çàïèñûâàþò â âèäå1f (t0 ) = √2πZ∞fb(x)eit0 x dx,−∞ïîäðàçóìåâàÿ ïðè ýòîì, ÷òî èíòåãðàë ïîíèìàåòñÿ â ñìûñëå ãëàâíîãî çíà÷åíèÿ, êàê ýòî óêàçàíî â ôîðìóëèðîâêå òåîðåìû.Ñóùåñòâåííóþ ðîëü â ïðèëîæåíèÿõ ïðåîáðàçîâàíèÿ Ôóðüå ê äèôôåðåíöèàëüíûì óðàâíåíèÿì èãðàåò ñëåäóþùàÿ òåîðåìà î ïðåîáðàçîâàíèèÔóðüå ïðîèçâîäíîé.Òåîðåìà 4.
Ïóñòü f (t) íåïðåðûâíî äèôôåðåíöèðóåìàÿ ôóíêöèÿ è ïðèíàäëåæèò âìåñòå ñî ñâîåé ïðîèçâîäíîé êëàññó L1 (R). Òîãäà ñïðàâåäëèâàôîðìóëà(Ff 0 )(x) = ix(Ff )(x).Åñòåñòâåííûì îáîáùåíèåì äàííîãî ðåçóëüòàòà ÿâëÿåòñÿÑëåäñòâèå 1. Ïóñòü f (t) n-ðàç íåïðåðûâíî äèôôåðåíöèðóåìàÿ ôóíêöèÿ è ïðèíàäëåæèò âìåñòå ñî ñâîèìè ïðîèçâîäíûìè êëàññó L1 (R). Òîãäàñïðàâåäëèâû ôîðìóëû(Ff (k) )(x) = (ix)k (Ff )(x), k = 1, ...n.(7.1)Ôîðìóëà 7.1 è ëåììà Ðèìàíà-Ëåáåãà ïîçâîëÿþò îïðåäåëèòü ïîâåäåíèå ïðåîáðàçîâàíèÿ Ôóðüå íà áåñêîíå÷íîñòè.7.1. Îñíîâíûå òåîðåìû131Ñëåäñòâèå 2. Ïóñòü ôóíêöèÿ f (t) óäîâëåòâîðÿåò óñëîâèÿì ñëåäñòâèÿ 1.Òîãäà(Ff )(x) = o(1/xn ) ïðè |x| → ∞.Îäíèì èç âàæíûõ êëàññîâ ôóíêöèé, â êîòîðîì ðàññìàòðèâàåòñÿ ïðåîáðàçîâàíèå Ôóðüå, ÿâëÿåòñÿ êëàññ S(R) áåñêîíå÷íî äèôôåðåíöèðóåìûõáûñòðî óáûâàþùèõ ôóíêöèé.
Ôóíêöèÿ f (t) ïðèíàäëåæèò êëàññó S(R), åñëèpn,m (f ) = sup |tm f (n) (t)| < ∞täëÿ âñåõ n, m. ïðîñòðàíñòâå S(R) ââîäèòñÿ ïîíÿòèå ñõîäèìîñòè. Ãîâîðÿò, ÷òî ïîñëåäîâàòåëüíîñòü {fk } ñõîäèòñÿ â ïðîñòðàíñòâå S(R) ê ôóíêöèè f , åñëèpn,m (fk − f ) → 0ïðè k → ∞ äëÿ âñåõ n, m.Òåîðåìà 5. Ïðåîáðàçîâàíèå Ôóðüå F âçàèìíî îäíîçíà÷íî è âçàèìíîíåïðåðûâíî îòîáðàæàåò ïðîñòðàíñòâî S(R) íà ñåáÿ.
Îáðàòíîå ïðåîáðàçîâàíèå Ôóðüå îïðåäåëÿåòñÿ ôîðìóëîé1(F −1 g)(t) = √2πZ+∞g(t)eitx dt.−∞Çàìåòèì, ÷òî âçàèìíàÿ íåïðåðûâíîñòü îçíà÷àåò, ÷òî íåïðåðûâíû îáàîòîáðàæåíèÿ F è F −1 , ò.å. åñëè fn → f â ïðîñòðàíñòâå S(R), òî F ±1 fn →F ±1 f.Çàäà÷à 7.1. Âûÿñíèòü êàêèå èç ïðèâåäåííûõ ôóíêöèé ïðèíàäëåæàò êëàññó S(R):22tn e−t , χ[0,+∞) (t)e−t , e−|t| , (1 + t2 )−1 , th t,dth t.dtÇàäà÷à 7.2. Äîêàçàòü, ÷òî ïðîñòðàíñòâî S(R) èíâàðèàíòíî îòíîñèòåëüíîîïåðàöèé äèôôåðåíöèðîâàíèÿ è óìíîæåíèÿ íà ìíîãî÷ëåíû.7.2.
Ïðèìåðû âû÷èñëåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå1327.2 Ïðèìåðû âû÷èñëåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå ïðîñòûõ ñëó÷àÿõ ïðåîáðàçîâàíèå Ôóðüå ìîæåò áûòü âû÷èñëåíî íåïîñðåäñòâåííûì èíòåãðèðîâàíèåì.Çàäà÷à 7.3. Íàéòè ïðåîáðàçîâàíèå Ôóðüå äëÿ ñëåäóþùèõ ôóíêöèé1. χ[a,b] (t),3. χ(−∞,0] (t) et ,2. e−α|t| ,4. χ[a,b] (t) sin t.Âî ìíîãèõ ñëó÷àÿõ ìîæíî èñïîëüçîâàòü ìîùíûå ìåòîäû òåîðèè àíàëèòè÷åñêèõ ôóíêöèé, â ÷àñòíîñòè, òåîðèþ âû÷åòîâ.Ïðèìåð 7.1.
Íàéòè ïðåîáðàçîâàíèå Ôóðüå äëÿ f (t) = 1/(t2 + 1).Ðåøåíèå. Ôóíêöèÿ f (z) = 1/(z 2 + 1) àíàëèòè÷íà â êîìïëåêñíîé ïëîñêîñòè çà èñêëþ÷åíèåì ïðîñòûõ ïîëþñîâ â òî÷êàõ z = ±i. Äëÿ âû÷èñëåíèÿïðåîáðàçîâàíèÿ Ôóðüå1F (x) = √2πZ+∞−∞e−itxdtt2 + 1ïîñòóïàåì ñëåäóþùèì îáðàçîì.Ðàññìîòðèì èíòåãðàëûJR± (x)1=√2πZe−izxdz,z2 + 1L±Rãäå L±R çàìêíóòûé êîíòóð, ïîëó÷àþùèéñÿ äîáàâëåíèåì ê îòðåçêó [−R, R]ïîëóîêðóæíîñòè CR± ñ öåíòðîì â íóëå ðàäèóñà R, ïðè÷åì CR+ ðàñïîëîæåíàâ âåðõíåé ïîëóïëîñêîñòè, à CR− â íèæíåé ïîëóïëîñêîñòè. Çàìåòèì, ÷òî−L+R ïîëîæèòåëüíî îðèåíòèðîâàííàÿ êðèâàÿ, à LR îòðèöàòåëüíî îðèåíòèðîâàíà.
Äàííûå èíòåãðàëû ëåãêî âû÷èñëÿþòñÿ ìåòîäîì âû÷åòîâ.Íàïîìíèì îñíîâíóþ òåîðåìó î âû÷åòàõ.Òåîðåìà 6. Ïóñòü g(z) ôóíêöèÿ, àíàëèòè÷åñêàÿ â îáëàñòè D çà èñêëþ÷åíèåì èçîëèðîâàííûõ ïîëþñîâ, è L çàìêíóòàÿ êóñî÷íî ãëàäêàÿ ïîëîæèòåëüíî îðèåíòèðîâàííàÿ êðèâàÿ, ñîäåðæàùàÿ âíóòðè ñåáÿ ïîëþñà7.2. Ïðèìåðû âû÷èñëåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå133z1 , z2 , ...zn . Òîãäà ñïðàâåäëèâà ôîðìóëàZnXg(z) dz = 2πiresz=zk g(z),k=1Lãäå resz=zk g(z) âû÷åò ôóíêöèè g(z) â ïîëþñå zk , êîòîðûé âû÷èñëÿåòñÿïî ôîðìóëå1dn−1resz=zk g(z) = lim(g(z)(z − zk )n ),n−1z→zk (n − 1)! dzãäå n ïîðÿäîê ïîëþñà zk . ðåçóëüòàòå ïðèìåíåíèÿ ýòîé òåîðåìû ïîëó÷àåìJR± (x)=pπ/2e±x .(7.2)Êàê, èñõîäÿ èç ýòîãî ðåçóëüòàòà, ïîëó÷èòü ïðåîáðàçîâàíèå Ôóðüå fb(x)?Ïåðåõîäÿ ê ïðåäåëó â ôîðìóëå 7.2, ïîëó÷àåì1fb(x) + lim √R→∞2πZpe−izxdz = π/2e±x .2z +1±CRßñíî, ÷òî äëÿ âû÷èñëåíèÿ fb(x) íóæíî çíàòü ïðåäåëû èíòåãðàëîâ ïîïîëóîêðóæíîñòÿì CR± ïðè R → ∞.Èçâåñòíàÿ ëåììà Æîðäàíà óêàçûâàåò óñëîâèÿ, ïðè êîòîðûõ èíòåãðàëûòàêîãî òèïà ñòðåìÿòñÿ ê íóëþ.Òåîðåìà 7.
(Æîðäàí) Ïóñòümax |g(z)| → 0 (max+ |g(z)| → 0)−z∈CRz∈CRïðè R → ∞ è x > 0 (x < 0). ÒîãäàZlimg(z)e−izx dz = 0R→∞−CR limZR→∞+CRg(z)e−izx dz = 0 .7.2. Ïðèìåðû âû÷èñëåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå134Ñîãëàñíî ëåììå Æîðäàíà èìååìfb(x) =èfb(x) =pπ/2ex ïðè x < 0pπ/2e−x ïðè x > 0,÷òî ìîæíî ïðåäñòàâèòü îäíîé ôîðìóëîéfb(x) =pπ/2e−|x| .Àíàëîãè÷íûå ñîîáðàæåíèÿ ìîæíî ïðèìåíÿòü â áîëåå ñëîæíîé ñèòóàöèè, êîãäà ó ïîäûíòåãðàëüíîé ôóíêöèè ìîæåò áûòü ñ÷åòíîå ÷èñëî ïîëþñîâ.Ïðèìåð 7.2. Íàéòè ïðåîáðàçîâàíèå Ôóðüå äëÿ ôóíêöèè f (t) = 1/ch t.Ðåøåíèå. Ôóíêöèÿ 1/ch z èìååò ïðîñòûå ïîëþñà â òî÷êàõ zk = i(k +1/2)π k = 0, ±1, ...
Ñîîòâåòñòâóþùèå âû÷åòû èìåþò âèä2πiresz=zk {e−izxe−izk x=} = 2πich zsh zke(k+1/2)πx= 2π= 2π(−1)k e(k+1/2)πx .sin (k + 1/2)xÇàìåòèì, ÷òî ïðè âû÷èñëåíèè èíòåãðàëîâ ïðåäåëüíûì ïåðåõîäîì ñ ïîìîùüþ òåîðèè âû÷åòîâ, âîîáùå ãîâîðÿ, íå îáÿçàòåëüíî îñóùåñòâëÿòü çàìûêàíèå îòðåçêà ñ ïîìîùüþ ïîëóîêðóæíîñòåé. Äëÿ ýòîé öåëè ìîæíî èñïîëüçîâàòü ñèñòåìó êîíòóðîâ Γ±n , óõîäÿùèõ íà áåñêîíå÷íîñòü ñîîòâåòñòâåííî ââåðõíåé èëè íèæíåé ïîëóïëîñêîñòè.Ïóñòü, íàïðèìåð, x < 0. Òîãäà ðàññìîòðèì çàìêíóòûé êîíòóð L+R,n ,ÿâëÿþùèéñÿ ãðàíèöåé ïðÿìîóãîëüíèêà ñ âåðøèíàìè −R, R, R+inπ, −R+inπ è èíòåãðàë+JR,n1=√2πZe−izxdz.ch zL+R,n+ äàííîì ñëó÷àå Γ+R,n = LR,n \ [−R, R]. Îñòàåòñÿ ïîêàçàòü, ÷òîZΓ+R,ne−izxdz → 0 ïðè R, n → ∞.ch z7.2.
Ïðèìåðû âû÷èñëåíèÿ ïðåîáðàçîâàíèÿ Ôóðüå135Äîêàæèòå ýòîò ôàêò ñàìîñòîÿòåëüíî. Ïîñëå ýòîãî ïîëó÷àåì, ÷òî ïðèx<0∞X√fb(x) =2π(−1)n e(n+1/2)πx .n=0Ñóììèðóÿ äàííûé ðÿä êàê ãåîìåòðè÷åñêóþ ïðîãðåññèþ, èìååì ïðè x < 0fb(x) =√2πpeπx/21π/2=.1 + eπxch πx/2Äîêàæèòå íåïîñðåäñòâåííûì âû÷èñëåíèåì, ÷òî ôîðìóëà âåðíà è äëÿ ñëó÷àÿ x > 0.Çàäà÷à 7.4. Íàéòè ïðåîáðàçîâàíèå Ôóðüå äëÿ ñëåäóþùèõ ôóíêöèét/sh t, sin t/t.Ïðèìåð 7.3. Íàéòè ïðåîáðàçîâàíèå Ôóðüå ôóíêöèè f (t) = e−αt2Ðåøåíèå.21 ñïîñîá.
Ôóíêöèÿ e−αz àíàëèòè÷åñêàÿ íà âñåé êîìïëåêñíîé ïëîñêîñòè.ÏîýòîìóZ2e−αz e−izx dz = 0,CRãäå CR ãðàíèöà ïðÿìîóãîëüíèêà ñ âåðøèíàìè â òî÷êàõ −R, R, R +iy, −R + iy . Èíòåãðàëû±R+iyZ2e−αz e−izx dz±Rñòðåìÿòñÿ ê íóëþ ïðè R → +∞. Ïîýòîìó1fb(x) = √2π+∞+iyZ2e−αz e−izx dz−∞+iyè íå çàâèñèò îò y .Îñóùåñòâëÿÿ çàìåíó z = t + iy , ñâåäåì èíòåãðàë íà âåùåñòâåííóþîñü è âûáåðåì y òàê, ÷òîáû ïîäûíòåãðàëüíàÿ ôóíêöèÿ ïðèíÿëà íàèáîëåå7.3.
Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííûõ ôóíêöèé136ïðîñòîé âèä. Èìååì1fb(x) = √2πZ+∞2e−α(t+iy) e−i(t+iy)x dt.−∞Ïîëàãàÿ y = −x/(2α), ïðèâåäåì èíòåãðàë ê âèäó1fb(x) = √2πZ+∞ 2x2e− 4α e−αt dt.−∞Çàìåíà ïåðåìåííîé τ = αt2 ïîçâîëÿåò ñâåñòè èíòåãðàë ê Γ ôóíêöèèÝéëåðà, â ðåçóëüòàòå îêîí÷àòåëüíî ïîëó÷àåìx21fb(x) = √ e− 4α .2α2 ñïîñîá.
Äèôôåðåíöèðóÿ èíòåãðàë1J(x) = √2πZ+∞2e−αt e−itx dt−∞è èíòåãðèðóÿ ïî ÷àñòÿì, ïðèõîäèì ê äèôôåðåíöèàëüíîìó óðàâíåíèþJ 0 (x) = −xJ(x).2αÏîýòîìóx2J(x) = Ce− 4α ,ãäå1c = J(0) = √2πZ+∞2e−αt dt.−∞Ñâîäÿ ïîñëåäíèé èíòåãðàë ê Γ-ôóíêöèè, óñòàíàâëèâàåì, ÷òî c =√1 .2α7.3 Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííûõ ôóíêöèéÍàïîìíèì îïðåäåëåíèå îáîáùåííûõ ôóíêöèé êëàññà S 0 (R).7.3. Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííûõ ôóíêöèé137Îïðåäåëåíèå 7.1.
Îáîáùåííîé ôóíêöèåé êëàññà S 0 (R) íàçûâàåòñÿ ëèíåéíûé íåïðåðûâíûé ôóíêöèîíàë f íà ïðîñòðàíñòâå îñíîâíûõ ôóíêöèéS(R).Äàëåå ïîìèìî îáû÷íîãî îáîçíà÷åíèÿ f (ϕ) äëÿ çíà÷åíèÿ ôóíêöèîíàëàíà îñíîâíîé ôóíêöèè áóäåì èñïîëüçîâàòü òàêæå îáîçíà÷åíèå < f, ϕ > .Îïðåäåëåíèå 7.2. Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííîé ôóíêöèè f ∈ S 0 (R)åñòü îáîáùåííàÿ ôóíêöèÿ, îïðåäåëÿåìàÿ ôîðìóëîé< Ff, ϕ >=< f, Fϕ >,ãäå ϕ ëþáàÿ îñíîâíàÿ ôóíêöèÿ.Äëÿ ïðåîáðàçîâàíèÿ Ôóðüå îáîáùåííûõ ôóíêöèé êàê è äëÿ îáû÷íûõôóíêöèé èñïîëüçóåòñÿ îáîçíà÷åíèå fb. Àíàëîãè÷íî îïðåäåëÿåòñÿ F −1 f :< F −1 f, ϕ >=< f, F −1 ϕ > .Ñïðàâåäëèâà ñëåäóþùàÿ òåîðåìà î ïðåîáðàçîâàíèè Ôóðüå â êëàññåS 0 (R).Òåîðåìà 8.
Ïðåîáðàçîâàíèå ÔóðüåF : S 0 (R) → S 0 (R)âçàèìíî îäíîçíà÷íî îòîáðàæàåò S 0 (R) íà S 0 (R). Åãî îáðàòíûì îòîáðàæåíèåì ÿâëÿåòñÿ F −1 .Òåîðåìà äîêàçûâàåòñÿ óäèâèòåëüíî ïðîñòî. Äëÿ ëþáîé îáîáùåííîéôóíêöèè f ∈ S 0 (R) ñïðàâåäëèâà öåïî÷êà ðàâåíñòâ< F −1 Ff, ϕ >=< Ff, F −1 ϕ >=< f, FF −1 ϕ >=< f, ϕ > .Ñëåäîâàòåëüíî,F −1 F = id.Àíàëîãè÷íî äîêàçûâàåòñÿ, ÷òîFF −1 = id.Ïðèâåäåì íåêîòîðûå ïðèìåðû âû÷èñëåíèÿ ïðåîáðàçîâàíèÿ Ôóðüåîáîáùåííûõ ôóíêöèé.7.3. Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííûõ ôóíêöèé138Ïðèìåð 7.4. Íàéòè ïðåîáðàçîâàíèå Ôóðüå δ -ôóíêöèè Äèðàêà.Ðåøåíèå. Íàïîìíèì, ÷òî δ -ôóíêöèÿ îïðåäåëÿåòñÿ ôîðìóëîé< δ, ϕ >= ϕ(0).Ïîýòîìó1< Fδ, ϕ >=< δ, Fϕ >= √2πÑëåäîâàòåëüíî, Fδ =Z+∞1ϕ(t) dt =< √ , ϕ > .2π−∞√1 .2πÏðèìåð 7.5.
Íàéòè ïðåîáðàçîâàíèå Ôóðüå ïðîèçâîäíîé δ -ôóíêöèè.Ðåøåíèå. Òàê êàê δ 0 (x) îïðåäåëÿåòñÿ ñîîòíîøåíèåì< δ 0 , ϕ >= −ϕ0 (0),òî1< Fδ 0 , ϕ >=< δ 0 , Fϕ >= − √2πÒàêèì îáðàçîì,Z+∞it(−it)ϕ(t) dt =< √ , ϕ > .2π−∞itFδ 0 = √ .2πÇàäà÷à 7.5. Äîêàçàòü ôîðìóëóFδ(n)in tn=√ .2πÇàäà÷à 7.6. Íàéòè ïðåîáðàçîâàíèå Ôóðüå äëÿ ñìåùåííîé δ -ôóíêöèè è ååïðîèçâîäíûõ. Ñìåùåííàÿ δ -ôóíêöèÿ îïðåäåëÿåòñÿ ôîðìóëîé< δh , ϕ >= ϕ(h).Ïðèìåð 7.6. Íàéòè ïðåîáðàçîâàíèå Ôóðüå îáîáùåííîé ôóíêöèè f = 1.7.3.
Ïðåîáðàçîâàíèå Ôóðüå îáîáùåííûõ ôóíêöèé139Ðåøåíèå. Ïðîùå âñåãî âîñïîëüçîâàòüñÿ ðåçóëüòàòàìè ïðèìåðà 7.4, ãäåóñòàíîâëåíî, ÷òî Fδ =÷òî F −1 δ =√1 .2π√1 .2πÀíàëîãè÷íî ýòîìó ïðèìåðó óñòàíàâëèâàåòñÿ,Òîãäà â ñèëó òåîðåìû îáðàùåíèÿ F1 =√2πδ.Îäíàêî ïðåäñòàâëÿåò èíòåðåñ ïðÿìîå äîêàçàòåëüñòâî ýòîé ôîðìóëû.Èìååì ñëåäóþùåå ñîîòíîøåíèå1< F1, ϕ >=< 1, Fϕ >= √2π1= lim √R→∞2πZ+RZ+∞Z+∞dtϕ(x)e−itx dx =−∞−∞r Z+∞Z+∞2sin Rxdtϕ(x)e−itx dx = limϕ(x)dx.R→+∞πx−R−∞−∞Äëÿ ïðåäåëüíîãî ïåðåõîäà âîñïîëüçóåìñÿ èçâåñòíûì èíòåãðàëîì ÄèðèõëåZ+∞−∞sin Rxdx = π.xÒîãäà ñïðàâåäëèâî ïðåäñòàâëåíèå< F1, ϕ >=√r2πϕ(0) + limR→+∞2πZ+∞sin Rx(ϕ(x) − ϕ(0))dx.x−∞Òàêèì îáðàçîì, îñòàåòñÿ ïîêàçàòü, ÷òî ïðåäåë èíòåãðàëà â ïîñëåäíåé ôîðìóëå ðàâåí íóëþ. Îáîçíà÷èì äàííûé èíòåãðàë ÷åðåç J(R) è ïðåäñòàâèì ââèäå ñóììû òðåõ èíòåãðàëîâ:J(R) = J1 (R) + J2 (R) + J3 (R),ãäåZJ1 (R) =(ϕ(x) − ϕ(0))|x|<δZJ2 (R) =ϕ(x)sin Rxdx,xsin Rxdx,x|x|>δZJ3 (R) = −|x|>δϕ(0)sin Rxdx.x7.4.
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