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The Second Mean Value Theorem for Complex Line Integral

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The Second Mean Value Theorem for Complex Line Integral 2018 INTERNATIONAL SCIENTIFIC CONFERENCE 16-17 November 2018, GABROVO THE SECOND MEAN VALUE THEOREM FOR COMPLEX LINE INTEGRAL Jelena Vujaković1, Stefan Panić1, Nataša Kontrec1 1 University of Priština, Faculty of Sciences and Mathematics Lole Ribara 29, Kosovska Mitrovica, Serbia [email protected]; [email protected]; [email protected]. Abstract In real iterations, several types of mean value theorems for definite integrals are used. In complex domain, we cannot specifically formulate the mean value theorem of a particular complex line integral fzdz , since we are unable to give L an appropriate geometric interpretation of the integral over the surface below a curve L (from z0 to z1 ). Based on the mean value theorems for a complex line integral in [Vujakovic J., The mean value theorem of line complex integral and Sturm function. Applied Mathematical Sciences 2014; 8 (37): 1817-1827.], we got the idea to formulate the second mean value theorem in complex domain for the product of two analytic functions. Keywords: mean value theorem, analytic function, power series, iteration. INTRODUCTION AND PRELIMINARIES fzdz 0. (3) In real iteration several types of mean value L theorems for definite integrals are used. We If z is interior point of contour L then will mention only two. Cauchy’s integral formula is valid Theorem 1. (The first mean value theorem 1 f [1]). Let fab:[ , ] be a continuous fz() d . (4) 2iz function. Then there exists (,)ab such that L b If fz is analytical on L , then L can be fxdxf ()( ba ). (1) replaced by the simplest closed contour, circle a KKzR (, ), so standard formula (4) is also Theorem 2. (The second mean value theorem valid for LK . [2]) If fab:[ , ] is a continuous function The mean value of function was obtained from on [,]ab with xab(,), mfxM() and (4), for zRei , 02, according to gab:[ , ] is an integrable function, then the following formula there exists in (,)ab such that 1 2 bb fz() fz ( Redi ) . (5) 2 fxgxdxf() () gxdx () . (2) 0 aaHowever, in solving complex differential It is usual to use the term of mean values for equations and for the needs of iterations, we complex integral and closed contours, could not use the formulas from (3) to (5). It according to which fz is analytical or was necessary to find a simple formula of mean value, as in a real case [4], where is a continuous while inside contour it can be certain kind of mean value between points z discontinuous and even non-analytical [3] This 0 was suggested by Cauchy’s fundamental and z , and az() is an analytic function. In [5] theorem we have shown that formula 442 International Scientific Conference “UNITECH 2018” – Gabrovo z Based on the theorems 1 and 2, together with azdz() a ()( z z ) (6) 0 the theorems 3 and 4, we got the idea to Lz: 0 formulate the second mean value theorem for where az is analytical function in area G , complex line integrals of product of analytical where is placed curve L on which integration functions fz and gz . is conducted, z and z are initial and final 0 Theorem 5. For integrals of the type (10), limits of integral, is certain point from G , where az is the analytical function in the does not have to be L but which satisfies certain standards of approximation, can be circle zR , the formula of the mean integral adopted as some kind of mean value for value is valid complex integral if integration path is open z line L . za z dz z2 a (11) In the further work we need the following L:0 statements. where is the interior point of the circle Theorem 3 ([5]). The integral of power zR. function az zn ,0 n , on Jordan curve L , Proof. For simplicity, let analytical function inside the circle zR can be replaced by fz az be given by power series integral on the shortest path, that is, on kk ( ) direction Oz , az azkk a01 az az . 12 z k 0 azdz() a () z (7) Since L:0 za z a zk1 where k k0 z (8) az az23 az azk 1 , n n 1 01 2k is the midpoint on the path and for the sake of analyticity, this series can be n integrated term by term. Hence, we obtain zz aa() ( ) . (9) 2 ak k n n 1 n 1 series zz. k 2 Formula (7) is the mean value formula for k 0 On the other hand, using the mean value complex integral. Theorem 3, that is formula (7), we obtain Theorem 4. [5] For arbitrary polynomial n equality Pz azk applies formula (7), for mean a nk zzza22k kk . (13) k 0 k 2 k value of integral, where is certain mean kk00 value of z with module lesser than zR and Now, we try to find the relationship between which depends on z , that is on Pzn . the mean value and the argument z , for each coefficient ak . From equality (13) MAIN RESULTS 2 k 2 aaz0 az12 az k z For the needs of iterations, for solving a 23 4k 2 complex differential equation, we found 22k important to find the complex line integral on za01 a a 2 ak the open path follows z aazaz az2 k Izazdz (10) 0 122 k k aa01,, a 2 ,,, ak L:0 23 4k 2 zz z where fz az and gz z. that is a0 0, . 32 k k 2 International Scientific Conference “UNITECH 2018” – Gabrovo 443 Since this is contradictory, we must observe Note that the last property also applies to a n the sums of analytic series k zk and multimorphic analytical functions zz,, k 0 k 2 lnzz , Artanh ... which are to a certain extent k unambiguous because the mean values are ak . If we evaluate these series, we see k 0 inside the circle zR. that the coefficients of the modular series For iterations it is also important to find a a k z k are smaller than the coefficient of z formula for the integral fzgzdz, of k 0 k 2 L:0 k series ak . The same can be expected the product of two analytical functions. k 0 Assume that az f zgz . Then, for the sum, because the coefficients ak directly affect the radius of convergence. according to formula (7), we have z In order to maintain the equality of the fzgzdzf g z, (15) modules of n th partial sums, as in the proof of L:0 Theorem 4, we conclude that between and for zR. z there is a connection z , which was to be proven. However, if part of the integral in formula (15) Theorem 6. The mean value for integral is solved, for example integral gzdz , then z L k za zdzk, , where az is the we need some kind of Second mean value L:0 theorem for a complex integral. analytical function in the circle zR , Theorem 7 (The second mean value theorem ( R ) is given by for a complex integral) For an integral of the product of two analytical functions fz and z zakk zdz z1 a . (14) gz , along the open path L which connects L:0 points z0 0 and z and which is contained in Proof. From the development of analytical the circle of radius zR valid formula function az in power series (12) we have z z1 z fzgzdzf gzdz zak zdz LL:0 :0 L:0 (16) z2 2 n gfzdz , k 1 aaz0 az12 az n z L:0 kk123 k kn 1 7 where the modules ,,zz12 are inside the zakn1 . n circle zR . n0 Dividing by z k 1 0 and equating multipliers Proof. If in (15) we perform a grouping in the following way with ak , we have a0 0, and z zz z fzgzdzf g z, z . k 2 k 3 n kn1 L:0 Since this is contradictory, by repeating the in brackets, we again have the case from procedure as in the proof of Theorem 5, we Theorem 3, but with some which is different conclude that formula (14) is valid for from z . Without loss of generality, we can zR. assume that fz 1. Then 444 International Scientific Conference “UNITECH 2018” – Gabrovo z1 zR. gz gzdzz, , 1 L:0 Based on the previous case, we have zz zz so azdz22 azdz LL:0 :0 LL :0 :0 z z1 zz z2 fzgzdzf gzdz aza dz2. LL:0 :0 LL:0 :0 2! where , zzR. Analogously, if in (15) If we do not change the upper limit z of 1 integral, then is constant, a() is also we make a grouping constant and according to the second z mean value theorem, for , zR fzgzdzg f z, z L:0 we get zz z2 we again have in brackets structure of the aza dz2 mean value for integral, but now to some point LL:0 :0 2! zz 2 z2 for which also applies z aazdz 2 z z2 LL:0 :0 2! fzgzdzg fzdz , LL:0 :0 zzz2 aa dz2 where zzR2 . 2! LL:0 :0 Note that the same is true for multiple products z4 aa. z 4! fzgzhzdzf g h z In this way, the double integrals in the 0 iterations are treated as in the real region, only z1 that the arguments of the mean values are not f gzhzdz on the segment [0,x ], but are in some circle 0 . z2 zR fg hzdz 0 fghz ... 2 REFERENCE [1] Rudin W.
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