Конспект лекций по математическому анализу. Шерстнев А.Н. - 350 стр.

UptoLike

ВУЗ: 

КФУ | Казань

Составители: 

Рубрика: 

           Z                 Z
oTS@DA jf , f j d = (f , f ) d = 0, TO ESTX (SM. 207.14) f = f = f P.W.
                                 Z        Z            Z1
NA [0; 1]. tAKIM OBRAZOM, f d = f d = (R) 0 f (x) dx. sFORMULIRUEM
POLU^ENNYJ REZULXTAT.
     2. eSLI f INTEGRIRUEMA NA OTREZKE PO rIMANU, TO ONA INTEGRIRUEMA
PO lEBEGU I SOOTWETSTWU@]IE INTEGRALY SOWPADA@T.
     z A M E ^ A N I Q. 3. nEOGRANI^ENNYE FUNKCII WOOB]E NE INTEGRIRUEMY
PO rIMANU, NO NEKOTORYE IZ NIH INTEGRIRUEMY PO lEBEGU. nAPRIMER,
                                    ( ,1=2
                          f (x) = x0; ; ESLI       0 < x  1,
                                              ESLI x = 0,
NE INTEGRIRUEMA PO rIMANU. oDNAKO, f INTEGRIRUEMA PO lEBEGU. dEJST-
WITELXNO, POLOVIM fn(x) = x,1=2[n,2;1] (x) (n = 1; 2; : : :). qSNO, ^TO fn ! f
I PO P. 2             Z                Z1
                        fn d = (R) 0 fn (x) dx = 2 , n2  2:
oSTAETSQ WOSPOLXZOWATXSQ
                        Z 1 TEOREMOJ fATU.
     4. eSLI lim (R) jf (x)j dx < +1 , TO f INTEGRIRUEMA PO lEBEGU NA
               "!Z0+      "           Z1
[0; 1], PRI^EM f d = "lim        (R) f (x) dx (!!).
                        Z 1 !0+ "
     5. eSLI lim (R) jf (x)j dx = +1, TO f NE INTEGRIRUEMA PO lEBEGU,
               "!0+ Z "
                          1
DAVE ESLI "lim!0+
                   (R )
                        "
                            f (x) dx SU]ESTWUET.
 fpOLOVIM fn = f  (1=n;1] (n = 1; 2; : : :). tOGDA jfnj  jf j; jfnj P,!
                                                                         .W. jf j. eSLI
                                                                           Z
DOPUSTITX, ^TO f INTEGRIRUEMA PO lEBEGU, TO W SILU 207.12 jf j d <
+1; W ^ASTNOSTI,
                           Z1               Z           Z
                      (R) jf (x)j dx = jfnj d  jf j d;
                             1=n
              Z1
ODNAKO, (R) " jf (x)j dx ! +1 (" ! 0).g


                                         350

Страницы