Riemann Integral of Functions R into C

Transcription

1 FORMLIZED MTHEMTICS Vol. 18, No. 4, Pges , 2010 Riemnn Integrl of Functions R into C Keiichi Miyjim Ibrki University Fculty of Engineering Hitchi, Jpn Tkhiro Kto Grdute School of Ibrki University Fculty of Engineering Hitchi, Jpn Ysunri Shidm Shinshu University Ngno, Jpn Summry. In this rticle, we define the Riemnn Integrl on functions R into C nd proof the linerity of this opertor. Especilly, the Riemnn integrl of complex functions is constituted by the redefinition bout the Riemnn sum of complex numbers. Our method refers to the [19]. MML identifier: INTEGR16, version: The terminology nd nottion used here hve been introduced in the following rticles: [5], [1], [16], [18], [4], [6], [7], [15], [10], [13], [11], [12], [2], [3], [8], [17], [21], [9], [14], nd [20]. 1. Preliminries One cn prove the following proposition (1) For every complex number z nd for every rel number r holds R(r z) = r R(z) nd I(r z) = r I(z). Let S be finite sequence of elements of C. The functor R(S) yielding finite sequence of elements of R is defined s follows: (Def. 1) R(S) = R(S qu prtil function from N to C). The functor I(S) yields finite sequence of elements of R nd is defined s follows: (Def. 2) I(S) = I(S qu prtil function from N to C). 201 c 2010 University of Biłystok ISSN (p), (e)

2 202 keiichi miyjim et l. Let be closed-intervl subset of R, let f be function from into C, let S be non empty Division of, nd let D be n element of S. finite sequence of elements of C is sid to be middle volume of f nd D if it stisfies the conditions (Def. 3). (Def. 3)(i) len it = len D, nd (ii) for every nturl number i such tht i dom D there exists n element c of C such tht c rng(f divset(d, i)) nd it(i) = c vol(divset(d, i)). Let be closed-intervl subset of R, let f be function from into C, let S be non empty Division of, let D be n element of S, nd let F be middle volume of f nd D. The functor middle sum(f, F ) yields n element of C nd is defined by: (Def. 4) middle sum(f, F ) = F. Let be closed-intervl subset of R, let f be function from into C, nd let T be DivSequence of. function from N into C is sid to be middle volume sequence of f nd T if: (Def. 5) For every element k of N holds it(k) is middle volume of f nd T (k). Let be closed-intervl subset of R, let f be function from into C, let T be DivSequence of, let S be middle volume sequence of f nd T, nd let k be n element of N. Then S(k) is middle volume of f nd T (k). Let be closed-intervl subset of R, let f be function from into C, let T be DivSequence of, nd let S be middle volume sequence of f nd T. The functor middle sum(f, S) yields complex sequence nd is defined s follows: (Def. 6) For every element i of N holds (middle sum(f, S))(i) = middle sum(f, S(i)). 2. Definition of Riemnn Integrl of Functions R into C Next we stte two propositions: (2) For every prtil function f from R to C nd for every subset of R holds R(f ) = R(f). (3) For every prtil function f from R to C nd for every subset of R holds I(f ) = I(f). Let be closed-intervl subset of R nd let f be function from into C. Observe tht R(f) is qusi totl nd I(f) is qusi totl. We now stte severl propositions: (4) Let be closed-intervl subset of R, f be function from into C, s be non empty Division of, D be n element of s, nd S be middle volume of f nd D. Then R(S) is middle volume of R(f) nd D nd I(S) is middle volume of I(f) nd D.

3 Riemnn integrl of functions R (5) For every finite sequence F of elements of C nd for every element x of C holds R(F x ) = R(F ) R(x). (6) For every finite sequence F of elements of C nd for every element x of C holds I(F x ) = I(F ) I(x). (7) Let F be finite sequence of elements of C nd F 1 be finite sequence of elements of R. If F 1 = R(F ), then F 1 = R( F ). (8) Let F be finite sequence of elements of C nd F 2 be finite sequence of elements of R. If F 2 = I(F ), then F 2 = I( F ). (9) Let be closed-intervl subset of R, f be function from into C, S be non empty Division of, D be n element of S, F be middle volume of f nd D, nd F 1 be middle volume of R(f) nd D. If F 1 = R(F ), then R(middle sum(f, F )) = middle sum(r(f), F 1 ). (10) Let be closed-intervl subset of R, f be function from into C, S be non empty Division of, D be n element of S, F be middle volume of f nd D, nd F 2 be middle volume of I(f) nd D. If F 2 = I(F ), then I(middle sum(f, F )) = middle sum(i(f), F 2 ). Let be closed-intervl subset of R nd let f be function from into C. We sy tht f is integrble if nd only if: (Def. 7) R(f) is integrble nd I(f) is integrble. We now stte three propositions: (11) For every prtil function f from R to C holds f is bounded iff R(f) is bounded nd I(f) is bounded. (12) Let be non empty subset of R, f be prtil function from R to C, nd g be function from into C. If f = g, then R(f) = R(g) nd I(f) = I(g). (13) Let be closed-intervl subset of R nd f be function from into C. Then f is bounded if nd only if R(f) is bounded nd I(f) is bounded. Let be closed-intervl subset of R nd let f be function from into C. The functor integrl f yielding n element of C is defined s follows: (Def. 8) integrl f = integrl R(f) + integrl I(f) i. Next we stte two propositions: (14) Let be closed-intervl subset of R, f be function from into C, T be DivSequence of, nd S be middle volume sequence of f nd T. Suppose f is bounded nd integrble nd δ T is convergent nd lim(δ T ) = 0. Then middle sum(f, S) is convergent nd lim middle sum(f, S) = integrl f. (15) Let be closed-intervl subset of R nd f be function from into C. Suppose f is bounded. Then f is integrble if nd only if there exists n element I of C such tht for every DivSequence T of nd for every middle volume sequence S of f nd T such tht δ T is convergent nd lim(δ T ) = 0

4 204 keiichi miyjim et l. holds middle sum(f, S) is convergent nd lim middle sum(f, S) = I. Let be closed-intervl subset of R nd let f be prtil function from R to C. We sy tht f is integrble on if nd only if: (Def. 9) R(f) is integrble on nd I(f) is integrble on. Let be closed-intervl subset of R nd let f be prtil function from R to C. The functor f(x)dx yields n element of C nd is defined by: (Def. 10) f(x)dx = R(f)(x)dx + We now stte two propositions: I(f)(x)dx i. (16) Let be closed-intervl subset of R, f be prtil function from R to C, nd g be function from into C. Suppose f = g. Then f is integrble on if nd only if g is integrble. (17) Let be closed-intervl subset of R, f be prtil function from R to C, nd g be function from into C. If f = g, then f(x)dx = integrl g. Let, b be rel numbers nd let f be prtil function from R to C. The functor (Def. 11) b b f(x)dx yielding n element of C is defined by: f(x)dx = b b R(f)(x)dx + I(f)(x)dx i. 3. Linerity of the Integrtion Opertor Next we stte severl propositions: (18) Let c be complex number nd f be prtil function from R to C. Then R(c f) = R(c) R(f) I(c) I(f) nd I(c f) = R(c) I(f)+I(c) R(f). (19) Let be closed-intervl subset of R nd f 1, f 2 be prtil functions from R to C. Suppose f 1 is integrble on nd f 2 is integrble on nd dom f 1 nd dom f 2 nd f 1 is bounded nd f 2 is bounded. Then f 1 + f 2 is integrble on nd f 1 f 2 is integrble on nd (f 1 + f 2 )(x)dx = f 1 (x)dx + f 2 (x)dx nd (f 1 f 2 )(x)dx = f 1 (x)dx f 2 (x)dx. (20) Let r be rel number, be closed-intervl subset of R, nd f be prtil function from R to C. Suppose dom f nd f is integrble on

5 Riemnn integrl of functions R nd f is bounded. Then r f is integrble on nd (r f)(x)dx = r f(x)dx. (21) Let c be complex number, be closed-intervl subset of R, nd f be prtil function from R to C. Suppose dom f ndf is integrble on nd f is bounded. Then c f is integrble on nd (c f)(x)dx = c f(x)dx. (22) Let f be prtil function from R to C, be closed-intervl subset of b R, nd, b be rel numbers. If = [, b], then f(x)dx = f(x)dx. (23) Let f be prtil function from R to C, be closed-intervl subset of b R, nd, b be rel numbers. If = [b, ], then f(x)dx = f(x)dx. References [1] gnieszk Bnchowicz nd nn Winnick. Complex sequences. Formlized Mthemtics, 4(1): , [2] Grzegorz Bncerek. The fundmentl properties of nturl numbers. Formlized Mthemtics, 1(1):41 46, [3] Grzegorz Bncerek. The ordinl numbers. Formlized Mthemtics, 1(1):91 96, [4] Grzegorz Bncerek nd Krzysztof Hryniewiecki. Segments of nturl numbers nd finite sequences. Formlized Mthemtics, 1(1): , [5] Czesłw Byliński. The complex numbers. Formlized Mthemtics, 1(3): , [6] Czesłw Byliński. Functions nd their bsic properties. Formlized Mthemtics, 1(1):55 65, [7] Czesłw Byliński. Functions from set to set. Formlized Mthemtics, 1(1): , [8] Czesłw Byliński. Prtil functions. Formlized Mthemtics, 1(2): , [9] Czesłw Byliński. The sum nd product of finite sequences of rel numbers. Formlized Mthemtics, 1(4): , [10] Noboru Endou nd rtur Korniłowicz. The definition of the Riemnn definite integrl nd some relted lemms. Formlized Mthemtics, 8(1):93 102, [11] Noboru Endou, Ktsumi Wski, nd Ysunri Shidm. Drboux s theorem. Formlized Mthemtics, 9(1): , [12] Noboru Endou, Ktsumi Wski, nd Ysunri Shidm. Definition of integrbility for prtil functions from R to R nd integrbility for continuous functions. Formlized Mthemtics, 9(2): , [13] Noboru Endou, Ktsumi Wski, nd Ysunri Shidm. Sclr multiple of Riemnn definite integrl. Formlized Mthemtics, 9(1): , [14] Jrosłw Kotowicz. Convergent sequences nd the limit of sequences. Formlized Mthemtics, 1(2): , [15] Keiichi Miyjim nd Ysunri Shidm. Riemnn integrl of functions from R into R n. Formlized Mthemtics, 17(2): , 2009, doi: /v y. [16] dm Numowicz. Conjugte sequences, bounded complex sequences nd convergent complex sequences. Formlized Mthemtics, 6(2): , 1997.

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