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n A : H G H, G D(A) H R(A) = AD(A) G H = G A1 A2 D(A1) = D(A2) A1f = A2f f D(A1) A A A D() D(A) f = Af f D(A) A f D(A) > 0 > 0 : g D(A), f - g < Af - Ag < G f G A f, g D(A),, C A(f + g) = Af + Ag D H = C[0, 1], Df = f, D(D) = C D D D(D) = C1 L 1 H = D(L) = C[0, 1], (Lf)(x) = l(x, y)f(y)dy l(x, y) 0 A A-1 D(A-1) = R(A), R(A-1) = D(A) A-1Af = f f D(A); AA-1f = f f R(A) Af = h Af = 0 Af = 0 h x A : C[0, 1] C[0, 1], Af(x) = f(y)dy.

0 Af = 0 x f(y)dy 0 f(x) = 0 x 0 x0 : f(x0) = y f(x) : x (x0, x0 + ), |y - | q > 0 f(x) (y -, y + ) 0 < < x0+ x0 x0+ x0+ 0 = f(x)dx = f(x)dx + f(x)dx = f(x)dx |y - | > 0 0 0 x0 x0 Af = 0 f 0 h Af = h A C > 0 :

f D(A) Af C f inf(C) = A Af Af A = sup = sup = sup Af, f f fD(A) fD(A), f =p fD(A), f =1 p R \ {0} p f D(A) \ {O} g = f f Af Ag g D(A), g = p = f g f D(A) D(A) A f D(A) > 0 > 0 :

g D(A), f - g < Af - Ag < f D(A) g = f - f + f > 0 > 0 : f D(A), f - f < A(f - f ) < A(f - f ) = Af - Af A f f Af C f ; Af - Ay = A(f - y) C t - y < f - y < = C A f : f = 1, Af f A(g + ) - A(g) = A, = A. : 1 0, A > 1 X, R : x X x 1 x 2 x 1 X n 2 f = k, 1 n = dimX k k f f, k : k f Af n2 max |aij| f, i,j Af n2 maxi,j |aij| = const f Df = f D : C[a, b] C[a, b]; D(D) = C(a, b).

Df1 - Df2 = D(f1 - f2) D Df f1 = 1, f2 0. D f = e-nx f n D : C1[a, b] C[a, b], C1 f = maxt |f(t)| + maxt |f (t)| T D(T ) {{fn} D(T ) : lim fn = f, lim T fn = g} {f D(T ), g = T f} n n lim fn = f, lim hn = f, lim T fn = lim T hn.

n n n n lim fn = f, lim hn = f lim T fn = lim T hn n n n n lim fn = f f D(T ) / n T T {fn} : fn D(T )} T (f D(T ) \ D(T )) = lim T fn, f = lim fn.

n n C[0, 1]; f 2 = f2(t)dt (Af)(t) = tf(1).

C 1 1 2 Af = t2f2(1)dt = f2(1) f C. Af1 - Af2 = (f1 (1) - f2 (1)) 3 f1 - f2 = lim f(1) - lim f(2) = n n n n f(1) 0 f1 n n 0 t [0, 1 - ] 0 t [0, 1) (2) n fn = f2 = 1 n(t - 1 + ) t [1 -, 1] 1 t = n n Af1 - Af2 = = T, D(T ) H. g D(T ), g : f D(T )(T f, g) = (f, g) g = T g T T A f, g D(A) : (Af, g) = (f, Ag);

D(A) H A A = A D(A) = D(A) Af = Af D(A) D(A) b H = 2[a, b]; Af = f ; (f, g) = f(t)g(t)dt.

a b b b (Af, g) = f (t)g(t)dt = fg|b - f(t)g (t)dt = - f(t)g(t)dt = (f, Ag) a a a a D(A) = {f : f(a) = f(b) = 0}) Ag = -g = Ag Df = if b b b (Df, g) = i f (t)g(t)dt = ifg|b -i f(t)g (t)dt = f(t)ig (t)dt = (f, Dg) a a a a D(D) = {f : f(a) = f(b) = 0}.

D(D) f(a)g(a) - f(b)g(b) = f(a) g(b) f(b) = f g, f(a) = 0 f(a) = f(b) = f(b) f(b) g(a) f(a) f(a)f(a) = f(b)f(b) |f(a)| = |f(b)| || = D(D) = {f : f(a) = f(b), || = 1}.

H = 2 [0, ) ; Df = if.

(Df, g) = i f (t)g(t)dt = ifg| - i f(t)g (t)dt = f(t)ig (t)dt = (f, Dg) 0 0 D(D) = {f : f(0) = 0} f(0)g(0) = 0; f g |f(0)|2 = H = 2(-, ); Df = if.

A B D(A) B ! A : D() = B, = A f B, f D(A) {fn} D(A) : fn f f = / lim Afn n gn : gn f A(fn -gn) A fn -gn 0, A(fn - gn) 0 lim Agn = f n f Af C f. f = lim Afn lim A fn C lim fn = C f, n n n f C f, A.

A = max |Af|, = max |f|, A, f =1,fD(A) f =1,fB = A f : 1f = 2f f D(A) / f {fn} : fn f, n 1fn = 2fn lim 1fn = lim 2fn n n 1f = 2f A G(A) = {f, Af}.

G = {Af, f}.

M H H A {O, g} g = O A AO = O {f, g} {f, g } Af = g, Af = g {f, Af} + {f, Af } = {f +f, A(f +f )} A f +f = f Af = Af M {O, g} g = O M M H H {f, g} = ( f 2 + g 2) {fn, Afn} {f0, Af0} {fn, Afn} {f0, Af0} {fn - f0, A(fn - f0)} lim fn = flim Afn = Aflim Afn - Af0 A A A-1 A A- A D(A) = H A A = A Af 2 = (Af, Af) = (AAf, f) A Af f, Af A f A A A Af A f A = A G (-L) = G(L) G G(L) G(L) ({f, Lf} {-Lg, g}) = -(f, Lg) + (Lf, g) = L an G(L) an a a G(L) G(L) G (-L) {-Lf, f} {f, Lf} M M < 1 f - Mf = h h (I - M)-1 = Mn (I - M)-1- M n= N J = Mn, JN = Mn n=0 n=N JN(I - M) = (Mn - Mn+1) = I - MN+1.

n= 0 MN+1 M N+1 N 0 JN N I Mn = (I -M)-- n= Mn Mn M n n=0 n=0 n= (I - M)-1 = Mn Mn M n.

1 - M n=0 n=0 n= L L-L : L - L0 < L- L-L = L0 - (L0 - L) = L0(I - L-1(L0 - L)).

(I - L-1(L0 - L))-1 = (L-1(L0 - L))n 0 n= L-L-1 = (L-1(L0 - L))nL-1, L-1.

0 1 - L-1(L0 - L) n= R() = R = (A - I)-1 A p(A) x = 0 : Ax = x R = (A - I)- c(A) R = (A - I)- H r(A) R = (A - I)- H R = (A-I)- (A) (A) = p(A) c(A) r(A) A : > A def A A - I A + || I = A + || M = M < || A B (I - )-|| A 1 (I - M)-1 = I - )-1 (I - A)-1.

A || || - A 1 || A - I D(A) R(A) f D(A) : Af f 0 : (A - I)f A - I R - R = ( - )RR R - R = RR - RR R(I - R) = R(I - R)|(A - I)(A - I) (I - R)(A - I) = (I - R)(A - I) A - I - I = A - I - I RR = RR R+-R R+R lim = lim = lim R+R = R R 0 0 R An (A - I)-1 = || > A n+n= n r(A) = lim An n || r(A) r(A) A x C[0, 1]; (Af)(x) = f(t)dt f = max |f(t)| f : f = 1 f(t) 1 t [0, 1]. f t[0,1] x x Af = max | f(t)dt| max | 1dt| = 1.

x[0,1] x[0,1] 0 A 1 f 1 A = x s (Anf)(x) = f(t)dt dy 0 x (x-t)n- (Anf)(x) = f(t)dt [0, 1] (n-1)! x (1 - t)n-1 1 (1 - t)n An dt = - =.

(n - 1)! (n - 1)! n n! 1 n 1 e n n n An = ( 2n( )n)- n! e n r(A) = A R(A - I) = H R x : z D(A) : D(A) = H ((A - I)z, x) = 0 x R(A - I) (Az, x) - (z, x) = 0 (z, Ax) - (z, x) = 0 (z, (A - I)x) = z H Ax = x R = z : x(x, (A - I)z) = 0 x D(A) ((A - I)x, z) = z : x((A - I)x, z) = 0 z R(A - I) H = R(A - I) Ker(A - I) = + i, = (A - I)- (A - I)-((A - I)f, (A - I)f) = (Af, Af) - 2(Af, f) + ||2(f, f) = (Af, Af) - 2(Af, f) + 2(f, f) + 2(f, f) = ((A - I)f, (A - I)f) + 2(f, f) 2 f 1 (A - I)-1g (A - I)f = g g 2 2 (A - I)-1g 2 g (A - I)-1 D((A - I)-1) = R(A - I) R(A - I) = H R(A - I) (A - I)- (A - I)-1 R(A - I) = R(A - I) = H H R(A - I) = H R(A - I) = R(A - I) = H R(A - I) = H f(t) b K(s; t)f(t) dt = h(s) a b f(s) - K(s; t)f(t) dt = h(s) a K h b Mf = K(s; t)f(t) dt.



a f - Mf = h (1) M < b b b b M K(s; t) dt = || K(s; t) dt = || K(s; t) dt ds a a a a b b b b || |K(s; t)| dt ds || |K(s; t)|2 dtds a a a a b b || |K(s; t)|2 dsdt < a a (I - M)-1 = Mn, n f0 = h, f1 = h + Mh,... fn = Mkh b b M2h(s) = 2 K2(s; t)h(t) dt, K2(s; t) = K(s; t1)K(t1; t) dta a b b b Kn(s; t) = dt1 dt2... dtn-1 K(s; t1)K(t1; t2)... K(tn-1; t) a a a Kn(s; t) f = Mn h = h + n-1Knh = h + Rh M = K n=0 n=R R(s; t; ) b f(s) = h(s) + R(s; t; )h(t) dt a R(s; t; ) K(s; t) = st k K(s; t) = ai(s)bi(t) ai, bi b n n f(s) = ai(s) bi(t)f(t) dt + h(s) = h(s) + ciai(s) () 1 a b n n n h(s) + ciai(s) = ai(s) bi(t) h(t) + cjaj(t) dt + h(s) 1 1 a b b b n n ci = bi(t) h(t) + cjaj(t) dt = bi(t)h(t) dt + cj aj(t)bi(t) dt 1 a a a hi eijcj ci ci = hi + eijcj j E = c - h, (I - E) = h c c ci () K(s; t) = s2t + st b f(s) - K(s; t)f(t) dt = h(s) a [a, b] n n b - a f(tj) - K(tj; ti)f(ti) = h(tj) n i=D(s;t;) R(s; t; ) =, D() D() = 1 + (-1)nn dn n! n= b b K(t1; t1)... K(t1; tn) () dn = dt1... dtn.........

a a K(tn; t1)... K(tn; tn) D(s; t; ) = K(s; t) + (-1)n ndn(s;t) n! n= K(s; t) K(s; t1)... K(s; tn) b b K(t1; t) K(t1; t1)... K(t1; tn) dn(s; t) = dt1... dtn ()............

a a K(tn; t) K(tn; t1)... K(tn; tn) b R(s; t; ) = K(s; t) + R(s; t1; )K(t1; t) dta R(s; t; ) = n-1Kn(s; t) n= b b R(s; t1; )K(t1; t) dt1 = n-1Kn(s; t1)K(t1; t) dt1 = n=a a = n-1Kn+1(s; t) = n-1Kn(s; t) = n=1 n= 1 1 = n-1Kn(s; t) - K(s; t) = (R(s; t; ) - K(s; t)) n= D() n n n dn = V V = K2(tj; tk) Kmaxnn/j=1 k= n D() 1 + (-1)nn Kmaxnn/2 1 + (-1)nn nn/2 (e)n n n+n n! 2n ( ) n e n=1 n=1 n= dn(s; t) () D(s; t; ) = R(s; t; )D() b D(s; t; ) = K(s; t)D() + D(s; t1; )K(t1; t) dta D(s; t; ) = K(s; t) 1 + (-1)nn dn + n! n= b + K(t1; t) K(s; t1) + (-1)nn dn(s;t1) dtn! n=a K(s, t) + (-1)nn dn(s;t1) = n! n= K(s, t) + (-1)d1(s, t) + (-1)nn dn(s;t1) n! n= b b dn-1(s,t1) ndn D(s, t, ) = K(s, t) + K(s, t)(-1)d + + K(t1, t)K(s, t1)dt1 + n K(t1, t) 1 n! n! 2 a a D(s; t; ) b dn(s; t) = K(s; t)dn - n K(t1; t)dn-1(s; t1) dt1 (2) a dn(s; t) () () dn(s; t) dn(s; t) d1(s; t) = d1(s; t) dn(s; t) = dn(s; t) n b K(s; t) K(s; t1) d1(s; t) = dt1 = K(s; t)d1 - K2(s; t) K(t1; t) K(t1; t1) a dn(s; t) b b K(s; t) K(s; t1)... K(s; tn) K(t1; t) K(t1; t1)... K(t1; tn) dn(s; t) = dt1... dtn =............

a a K(tn; t) K(tn; t1)... K(tn; tn) K(s; t1) K(s; t2)... K(s; tn)............

n K(tk-1; t1) K(tk-1; t2)... K(tk-1; tn) = K(s; t)dn + (-1)k+2K(tk; t)............

k=1 K(tk+1; t1) K(tk+1; t2)... K(tk+1; tn)............

K(tn; t1) K(tn; t2)... K(tn; tn) k k = 2 t1 t2 t2 t k = k dn-1(s; t1) dn(s; t) K(s, t) - K b K(s, s) ds - , a b K(t, t1)K(s, s) ds - a b K(s, t1)K(t1, t) dt1 -.

a R(s, t, ) = + + + +....

- 2 +...

- - + + 2 - D(s, t, ) R(s, t, ) = =, D() 1 - + ( - ) +...

K - + 2- 3.

1 1 - 1 .

- + - - + 2 2 6 3 D() = b f(s) = h(s) + R(s, t, )h(t) dt a = 0 D(0) = b dn+1 = dn(s, s) ds D () a dn dn+D () = (-1)nn-1 = -d1 - (-1)nn = (n - 1)! n! n=1 n= b dn(s, s) ds b b dn(s, s) = -d1 - (-1)nn a = - K(s, s) ds - (-1)nn ds = n! n! n=1 n=a a b b dn(s, s) = - K(s, s) + (-1)nn ds = - D(s, s, ) ds, n! n=a a b (k) (k-1) D = - D (s, s, ) ds = 0.

a D(s,t,) 0 R(s, t, ) = D() R(s, t, ) (k-1) 0 k D() D(k)() = 0 D (s, s, ) = 0 (k - 1) r C-r(s, t) C-r+1(s, t) R(s, t, ) = + +..., C-r(s, t) = 0.

( - 0)r ( - 0)r- b b R(s, t1, )K(t1, t)dt1 = R(t1, t, )K(s, t1)dta a R(s, t1, ) = n-1Kn(s, t1) n= b b b Kn(s, t) = dt1 dt2... dtn-1K(s, t1)K(t1, t2)... K(tn-1, t) a a a b b Kn(s, t1)K(t1, t)dt1 = Kn(t1, t)K(s, t1)dta a s t C-r(s, t) b f(s) = 0 K(s, t)f(t) dt.

a b R(s, t, ) = K(s, t) + R(s, t1, )K(t1, t) dta b C-r(s, t) C-r(t1 - t) +... = K(s, t) - +... K(s, t1) dt( - 0)r ( - 0)r a b C-r(s, t) +... = K(s, t)( - 0)r + (C-r(t1, t) +...) K(s, t1) dt1.

a = b C-r(s, t) = 0 C-r(t1, t)K(s, t1) dt1, a b C-r(s, t) = 0 C-r(s, t1)K(t1, t) dt1.

a b (s) = 0 K(t, s)(t) dt a b f(s) = 0 K(s, t)f(t) dt a b f(s) = 0 K(s, t)f(t) dt a D() = Cr(s, t) D() = 0 f = Mf f - Mf = h, h = b f(s) = 0 + R(s, t, ) 0dt = a D().

b K(t, s)(t)dt = (t) a b fj(s) = 0 K(s, t)fj(t) dt a 0 f1(s), f2(s),..., fn(s) b fj(s) = K(s, t) fj(t) dt a fj(s) Cj = {fj} K(s, t) t Cnfn = f b |Cn|2 |f|2 ds a b 2 f (s) K(s, t) dt i i=a 2 f (s) fj(s) j = |0| b b b K(s, t) dt ds 1 fi(s)fi(s) ds = = +.

|0|2 |0|i=0 i=a a a b b K(s, t) dt = + a a < + n |0|i=0 f1(s), f2(s),..., fn(s) 0 1(s), 2(s),..., m(s) n n < m. K(s, t) L(s, t) = K(s, t) - j(s) fj(s) j= b f(s) = 0 L(s, t)f(t) dt () a b (s) = 0 L(t, s)(t) dt () a b f(t)fj(t) dt = 0 f () a b b n f(s) = 0 K(s, t)f(t) dt - 0 j(s) fj(t) f(t) dt j=a a b b b b b n K(s, t)k(s) ds f(t) dt - fj(t)f(t) dt j(s)k(s) ds = f(s)k(s) ds = j=a a a a a b b = f(t)k(t) dt - 0 fk(t)f(t) dt a a b fk(t)f(t) dt = a 0 () b f(s) = 0 K(s, t)f(t) dt f f(s) = a n Cifi(s) = 0 f(s)fi, i : Ci = 0. f i= m {j(s)} b m(s) = 0 K(t, s)m(t) dt a (s) = m(s) m > n () () () n < m n > m b f(s) = 0 K(s, t)f(t) dt a b f(s) = g(s)+0 K(s, t)f(t) dt.





a b g(s)(s) ds = 0, a b f(s) = g(s) + 0 s, t)f(t) dt a b b b b f(s)(s) ds = g(s)(s) ds + 0 f(t) (s)s, t) dt a a a a b b b f(s)(s) ds = g(s)(s) ds + f(t)(t) dt.

a a a b g(s)(s) ds = 0.

a b b b n f(s) = g(s) + 0 L(s, t)f(t) dt = g(s) + 0 s, t)f(t) dt - 0 j(s) fj(t) f(t) dt j=a a a b b b b b f(s)k(s) ds = 0 f(t) dt K(s, t)k(s) ds - 0 f(t)fk(t) dt k(s)k(s) ds.

a a a a a b f(t)fk(t) dt = 0, L a K.

b b (s) = 0 K(t, s)(t) dt g(s)(s) ds = 0.

a a K(s, t) = 1 + st a = 0, b = 1 1 f(s) = (1 + st)f(t) dt = f(t) dt + s tf(t) dt 0 0 f(s) = c1 + sc cc1 = f(s) ds = c1 + c1 cc2 = sf(s) ds = + 2 - = 0 = 8 2 13.

- 2 C1 C|L(s; t)| |K(s; t)| = 0 < < 1; L(s; t) - |s - t| s = t (s - t)K(s; t) X = C[0, 1] (f, g) = max |f -g| K C[0, 1] t[0,1] K s f(s) = h(s) + K(s; t)f(t) dt (1) a s h(s) = K(s; t)f(t) dt (2) a K(s; t) b f(s) = h(s) + L(s; t)f(t) dt a K(s; t), t s L(s; t) = 0, t < s L(s;t)(s-t) [0, 1] = L(s; t) L(s; t)(s - t) (s-t) [a, b] [a, b] L(s; t) |K1(s; t)| = |K(s; t)| C b s |K2(s; t)| = K1(s; t1)K(t1, t) dt1 = K(s; t1)K(t1; t) dt1 = a a s = K(s; t1)K(t1; t) dt C2(s - t) t s s s |K3(s; t)| = K(s; t1)K2(t1; t) dt1 |K(s; t1)|C2(t1 - t) dt C3 t1 dt1 a a t s -C2 t dt1 = C3 s2-t2 - C2t(s - t) C3 (s-t)2 t s s |K4(s; t)| = K(s; t1)K3(t1; t) dt1 C4 (t1-t)2 dt1 C4 (s-t) a 2 3! t (s - t)n-|Kn(s; t)| Cn (n - 1)! |R(s; t; )| C n-1Cn-1 (s-t)n-1 = CeC(s-t) (n-1)! n= D() K(s; t) s K(s; t) = K(s - t) f(s) = h(s) + K(s - t)f(t) dt (Lf)(p) = e-ptf(t) dt = f s L( K(s - t)f(t) dt)(p) = K f 1, t > L1(t) = 1(t) = p 0, t L(eat) = p-a a L sin at = p2+ap L cos at = a2+pn! Ltn = pn+ L(f(n)(t)) = pnf(p) - pn-1f(0) - pn-2f (0) - - fn-1(0) L(tf(t)) = -df(p) dp L(eatf(t)) = f(p - a) R(s; t; ) = R(s - t; ) [0, ) |f| Cebt b K h R K h h(p) f(p) = h(p) + K(p)f(p) f(p) = 1 - K(p) +i f(t) = f(p)ept dp, 2i -i f x f (x) - f(x) + (x - t)f (t) dt = x f(0) = - p(p) + 1 - f(p) + x(p(p) + 1) = x f f (p - 1 + )(p) = -f p 1 p1 = + i p 2 f(p) = p2-p+1 1 p2 = - i 2 f(t) = Res(p)ept f 1 2 -L-1 p = -L-1 p+ - = 1 p2-p+(p- )2+ 2 1 t 3 1 t 2 = -L-1 p- - L-1 = -e1/2t cos - sin 1 3 1 2 (p- )2+ (p- )2+ 2 4 2 K -aexp, |x| < a a2-x(x) = 0, |x| a K K f + K : f(x)(x) dx = (f, ), (f, ) K K (f, ) R K (, ) = (0) + dx (f1 + f2, ) = (f1, ) + (f2, ) (f, ) = (f, ) (x) C, f K : (f, ) = (f, ) K, C + + (x)f(x)(x) dx = f(x)((x)(x)) dx - K, C C / + + + f (x)(x) dx = (x)f(x) - (x)f(x) dx (x) - f (f, ) = -(f, ) 1, x (x) = 0, x < (x) + + (, ) = (x)(x) dx = (x) dx + (, ) = - (x) = - = -(() - (0)) = (0) (, ) = (, ) = (, ) = -(, ) = - (0) ((n), ) = (-1)n(n)(0) + (ln|x|, ) = ln|x|(x) dx = - + lim ln(-x)(x) dx + lim ln(x)(x) dx = 0 - + = lim ln(x)((x) + (-x)) dx (ln |x|, ), x - + + 1 1 (x) - (1 - x) def (1/x, ) = lim (x) dx + (x) dx = lim (x) dx 0 -x x x - (ln |x|, ) = (1/x, ) + + (ln |x|, ) = -(ln |x|, ) = ln|x| (x) dx = lim ( (x) + (-x)) = - + + = lim ln(x) (x) dx + ln(x) (-x) dx = + + + + ln(x)(x) - (x) dx - ln(x)(-x) + (-x) dx = = lim x x + ln()() - ln()(-) - -(-x) + (x) dx = = lim x () = (0) + (0) + O() - (-) = 2 (0) + O() () = = 1/ lim0 ( ln()) = lim0 ln() = lim0 -1/2 = lim0 = 1/ + (x) - (-x) def lim dx = (1/x, ) x ((f), ) = ( f + f, ) f f K, C ((f), ) = -(f, ) (f, ) = (f, ) (f, ) = ( f, ) (f, ) = (f, ) = -(f, + ) = -(f, ) - (f, ) ((f), ) = ( f, )+(f, ) = -(f, ) = (f, )-(f, + ) = ( f, )-(f, () ) = C ( f, ) + (f, ) = ( f, ) + (f, ) = ( f + f, ) x y + y = C y(x) = C = const x (1/x, ) + + lim x ((x) + (-x)) dx = lim ((x) + (-x)) dx = 0 x + - + = lim (x) dx - (x) dx = (x) dx - C y= x (xy, ) + (y, ) = -(y(x) ) + (y, ) = -(y, + xy ) + (y, ) = -(, x ) = -(xy, ) = + = -(C, ) = -C (x) dx = (x) -(x ) = -0 (0) = 0, (x) fn f : (fn, ) (f, ) fn f fn f (fn, ) (f, ) (fn, ) = -(fn, ) -(f, ) = (f, ) (fn, ) n (fn, ) (, ) 1 fn(x) = sin(n(x-x )), n N x-x [0; +) fn n sin(nx) n cos(nx) + + + 1 sin(nx)(x) dx = - cos(nx)(x) + cos(nx) (x) dx n n - = + + |cos(nx)| 1 1 const n | (x)| dx = n cos(nx) (x) dx n n - + + sin(x) sin(x) dx = 2 dx = x x - + e-axsin(x) dx =, a (0, +) a2 + + + + + e-ax da sin(x) dx = - e-ax sin(x) dx = x a=0 + + sin(x) = (0 - 1) - dx = = acrtg(a) = / x a2 + 1 (fn, ) lim fn = n+ + n+ fn(x)(x) dx (x0) C 1 sin(n(x - x0)) fn(x) =, n [0; +) x - x+ fn(x) dx = 1, n + 1 sin(n(x - x0)) (x0) = - (x0) dx x - x+ 1 sin(n(x - x0)) ((x) - (x0) dx 0 (1) x - x x0 [a; b] R \ [a; b] (x0) (x) - (x0) (x0) = lim xx0 - xx D : Dy = -y x [0; l], y(0) = y(l) = -y = y y + y = < 0 = -p2 - p2 = 0 = p y = A epx + B e-px A + B = 0, y(0) = A ep l + B e-p l = 0, y(l) = A = B = A = -B ep l = e-p l p = 0 < < > 0 = p2 + py = A cos(px) + B sin(px) A = 0, y(0) = 0 A =, sin(p l) = B sin(p l) = 0, y(l) = 0 B = p l = n, n Z 2np(D) = = l [0; +) y(l) = |y| < l, l + sin(px) p > -y = p2 y y1 = sin(p1 x) y2 = sin(p2 x) y1 y -y1 = p12 y1 y -y2 = p22 y2 y y1y2

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