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The Mean Value Theorem of Line Complex Integral and Sturm Function 1 Introduction and Preliminary Notes

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Applied Mathematical Sciences, Vol. 8, 2014, no. 37, 1817 - 1827 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4279 The Mean Value Theorem of Line Complex Integral and Sturm Function Jelena Vujakovi´c Faculty of Sciences and Mathematics, University of Priˇstina Lole Ribara 29, 38 220 Kosovska Mitrovica, Serbia Copyright c 2014 Jelena Vujakovi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract For linear complex differential equation of second order w (z)+a (z)=0, where a (z) is analytical function, two solutions w1,w2 are obtained by itera- tions (according to results from [4] where are present successive line integrals on unclosed arc L of integration path. If L would be contour, then due to analytics and Cauchy theorem every first integral in order in these successive integrals would be equal to zero, then all iterations are equal to zero. In this way we obtain only trivial solution. That is way it has tried to find simple mean value formula, as in real integral. Mathematics Subject Classification: 34E05, 34K11 Keywords: The mean value theorem, Iteration, Sturm function, Complex differential equation, Oscillation 1 Introduction and Preliminary Notes It is common that term mean value of complex integral has been introduced for closed contours (for the further details, see [2]), according to which f (z)is analytical or at least continuous, while inside contour can be discontinuous and even non-analitical. This was suggested by Cauchy’s fundamental theorem 1818 J. Vujakovi´c f (z) dz =0. (1.1) L f(ς) Let z is point from the interior of contour L. Then function ς−z is not analytical all over inside L, so Cauchy’s fundamental integral formula is in effect 1 f (ς) f (z)= dς . (1.2) 2πi ς − z L If f (z) is analytical on L , then L can be replaced by the simplest closed contour, circle K of radius R with center at z. So standard formula (1.2) also ≡ f(ς) applied for L K. Under mean value of function we consider value ς−z dς K divided with thelenght of circle arc S =2Rπ. Having in (1.2) that ς = z + R eiϕ, 0 ≤ ϕ ≤ 2π, we obtain 2π 1 f (z)= f z + Reiϕ dϕ . (1.3) 2π 0 If f (z)=u (x, y)+iv (x, y) is analytical, then u (x, y) and v (x, y) are harmonic functions, so for them theorem on mean value also applies 2π 2π 1 1 u (z )= u z + reiϕ dϕ , v (z )= v z + reiϕ dϕ . (1.4) 0 2π 0 0 2π 0 0 0 However, in solving ordinary differential equations in real area, formulas from (1.1) to (1.4) are of small importance because closed contour has not bee used. So, certain method for easier calculus or estimation of complex line integral has been searched for, and as well for real integral. For real integral, as is known, of great importance is first theorem on mean value of certain integral. That is why, for unclosed pathL, we need approximate value of integrals a (z) dz, a (z) dz2, za(z) dz2, whereat we shall examine L L L L L approximation by module. Therefore, the following hypothesis applies. The mean value theorem... 1819 Formula a (z) dz = a (ς)(z1 − z0) (1.5) L where 1. a (z) is analytical function in area G , where is placed curve L on which integration is conducted 2. z0 and z1 are initial and final limits of integral 3. ς is certain point from G, doesn’t have to be L, but which satisfies certain standards of approximation, first according to module and then according to argument. 4. A (z) is primitive function for a (z), which is possible to find by quadrature can be adopted as some kind of mean value of integral, if integration path is open line L. Fundaments for this hypothesis are: 1. If z1 = z0, that is if L exists and closes down, then we obtain a (z) dz = 0, which is fundamental formula of Cauchy. L 2. For real z ≡ x we have section L =[x0,x1] and formula (1.5) becomes x1 a (x) dx = a (ς)(x1 − x0) ,x0 ≤ ς ≤ x, and that is formula for mean value x0 of integral in real area. 3. If a (ς) is discontinuous in given area G where path L is placed, that f(ς) is a (ς)= ς−z , then Cauchy’s fundamental theorem or Cauchy’s theorem on residues are valid. 4. Besides that, also are in effect formulas (1.3) and (1.4) on mean value of analytical function a (z), that is, on its harmonic parts u =Rea (ς) ,v= Im a (ς). So, basic knowledge about analytics fit into assumption (1.5), but its construc- tion demands existence of singular points from a (z) which don’t have to be on L. Thus, if L bends and closes and if the length of its arc is S =2Rπ, then L becomes a circle K of length S, so in order to have first characteristic valid, ς must be out of circle K. 2 Main Results These are the main results of the paper. Theorem 2.1. For integral of power function a (z)=zn,n≥ 0 on arbitrary open curve, from z0 =0to point z1, theorem on mean value is applied 1820 J. Vujakovi´c z1 a (z) dz = a (z) dz = a (ς) z1, (2.1) 0 where z ς = √ 1 . (2.2) n n +1 Integral of each Jordan curve inside the circle |z|≤|z1| = R can be replaced by integral on shortest path, that is, on direction Oz1 and (2.1), where the mid point on the path was given with (2.2) and the mean value of the function is zn √z1 1 a (ς)=a n n+1 = n+1 . Proof. Let’s first observe the case n = 0, that is, let a a (z)=c is complex constant. Since the integral of analytic function does not depend on path L but only on primitive function A (z)=z for given a (z)=c, and from starting and final point, implies a (z) dz = cdz = c (z1 − z0). Here L L is c (z1 − z0)=a (ς)(z1 − z0), and that is, actually, formula (1.5) in which a (ς)=c and ς any point on segment z0z1. For n = 1, that is for function a (z)=z, implies a (z) dz = zdz = L L z2−z2 1 z2 − z2 a ς z − z a ς ς ς 1 1 0 z1+z0 2 ( 1 0 )= ( )( 1 0). Hence, since ( )= we have = 2 z1−z0 = 2 . Therefore, ς is arithmetic mean of final points z0 and z1. Let’s further review the case n =2.Ifa (z)=z2, then a (z) dz = L 2 1 3 − 3 1 3 − 3 − 1 2 z dz = 3 (z1 z0 ), that is 3 (z1 z0 )=a (ς)(z1 z0)ora (ς)= 3 (z1 + z0z1 L 2 2 2 +z0 ) = ς . From here, due to many-valued characteristics of squarer root we z2 z z z2 ± 1 + 0 1+ 0 2 have two values for ς = 3 . Notice that ς is arithmetic mean of 2 2 three squares z1 ,z0z1,z0 , taken in the starting and final point. n We are continuing with the procedure. Let a (z)=z . Then a (ς)= n 1 n n n−1 n √z1 ς = n z1 + z1 z0 + ... + z0 .Forz0 = 0 implies ς = n n . Since √ +1 1 +1 n n +1 → 1 for n →∞the mean value ς → z1, for n →∞, on segment Oz1, and a (ς) tends to a (z1). So, we obtained sequence {ς n} of points, z1 √z1 √z1 √z1 , , 3 , ..., n , which on segment Oz tends to z . On the other hand, 2 3 4 n+1 1 1 with induction it easy to prove that The mean value theorem... 1821 z1 cdz = cdz = cz1 = a (ς) z1 =⇒ a (ς)=c = const L 0 z1 z2 z zdz = zdz = 1 = a (ς) z = ςz =⇒ ς = 1 2 1 1 2 L 0 z1 3 2 2 z1 2 z1 z dz = z dz = = a (ς) z1 = ς z1 =⇒ ς = √ 3 3 L 0 z1 4 3 3 z1 3 z1 z dz = z dz = = a (ς) z1 = ς z1 =⇒ ς = √ 4 3 4 L 0 . z1 n+1 n n z1 n z1 z dz = z dz = = a (ς) z1 = ς z1 =⇒ ς = √ . n +1 n n +1 L 0 Let a (z) is arbitrary analytical function, defined by arbitrary convergent +∞ n power series a (z)= anz . There is a question is it formula (2.1) valid or n=0 some other variance of this formula, as well as where mid point ς is placed and what is the amount of mean value of a (ς). n k Theorem 2.2. For arbitrary polynomial Pn (z)= akz applies for- k=0 mula (2.1), for mean value of integral, where ς is certain mean value of mod- ule lesser than |z1| = R1, ( |ς| <R1), and which depends on z1, that is on Pn (ak,z). Proof. For simple case, when P1 (z)=a0 + a1z is linear function we have z1 2 z (2.1) P (z) dz = (a + a z) dz = a z + a 1 = P (ς) z 1 0 1 0 1 1 2 1 1 L 0 z1 for P1 (ς)=a0 + a1ς.
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