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 iterations (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

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

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 module lesser than |z1| = R1, ( |ς| 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. From here ς = 2 . So mean value exists and is exactly in the middle of segment z0z1 = Oz1. Besides that, ς is inside the circle of radius |z1| = R1. 1822 J. Vujakovi´c

From the integral of squared function

z1 2 3 z z (2.1) P (z) dz = a + a z + a z2 dz = a z + a 1 + a 1 = P (ς) z 2 0 1 2 0 1 1 2 2 3 2 1 L 0 we obtain

z z2 a ς + a ς2 = a 1 + a 1 . (2.3) 1 2 1 2 2 3

Here implies equality

z z2 F (a ,a ,ς)=a ς + a ς2 ,G(a ,a ,z )=a 1 + a 1 1 2 1 2 1 2 1 1 2 2 3

of two complex function with the same coeﬃcients a1,a2, but with diﬀer- ent complex arguments ς and z1. On the basis of equality (2.3) we obtain F (a1,a2,ς)=G (a1,a2,z1), and here implies the equality of their modules

|F (a1,a2,ς)| = |G (a1,a2,z1)| . (2.4)

In such way we can also estimate the maximum of modules, |F (a1,a2,ς)|≤ |z2| | || | | || 2| | |≤| | |z1| | | 1 a1 ς + a2 ς and G (a1,a2,z1) a1 2 + a2 3 . Here the limits are certain real positive numbers. Since the coeﬃcients of limits |a1| and |a2| are mutual, and since we don’t know which of max |F (a1,a2,ς)| and max |G (a1,a2,z1)| is larger limit, in order to maintain (2.5) it suﬃciently take, |z2| | | | | | | | | | 2| | | |z1| 1 in special case, that a1 = a2 . Then a1 ( ς + ς )= a1 2 + 3 , that |z | |z | is |ς| = 1 and |ς| = √1 . Since we have obtained the contradictory, formula 2 3 (2.4) can be sustainable only if |ς| < |z1|. Let’s check whether formula is applicable for cubic function. Integral of this function is

z1 . 2 3 (2 1) P3 (z) dz = a0 + a1z + a2z + a3z dz = P3 (ς) z1 L 0 The mean value theorem... 1823

2 3 4 z1 z1 z1 2 3 Here appears a0z1 + a1 2 + a2 3 + a3 4 =(a0 + a1ς + a2ς + a3ς ) z1, or after z2 z3 z1 1 1 2 3 cancellation with z1 =0,a1 2 + a2 3 + a3 4 = a1ς + a2ς + a3ς . We again obtain two diﬀerent complex functions with arguments z1 and ς, and with equal coeﬃcients z z2 z3 F (a ,a ,a ,ς)=a ς + a ς 2 + a ς3 ,G(a ,a ,a ,z )=a 1 + a 1 + a 1 . 1 2 3 1 2 3 1 2 3 1 1 2 2 3 3 4

From F (a1,a2,a3,ς)=G (a1,a2,a3,z1) implies the equality of modules, and from comparison of coeﬃcients, in order to have special case |a1| = |a2| = |a3| |z | |z | |z | 1 √1 √1 fulﬁlled, |ς| = , |ς| = , |ς| = 3 . Appears that, ς must be lesser than 2 3 4 |z | |z | |z | 1 √1 √1 , and then 3 . Therefore |ς| < |z |. 2 3 4 1 We are concluding that theorem applies for all forms of complex polynomial, because in general case if L is open arc of certain curve, we have

n n k+1 k z (2.1) Pn (z) dz = akz dz = ak = Pn (ς) z . k +1 1 L L k=0 k=0

From here, after cancellation with z1 = 0 we obtain polynomial equation

2 n+1 2 n z1 z1 z a + a ς + a ς + ···+ anς − a − a − a −···−an =0. 0 1 2 0 1 2 2 3 n +1

Let an = 0. Then according to fundamental theorem of Algebra, upper equation has exactly n diﬀerent roots, real and complex. If we form in the same way functions

2 n F (ai,ς)=a1ς + a2ς + ···+ anς , 2 n+1 z1 z1 z1 G (ai,z )=a + a + ···+ an ,i= 1,n. 1 1 2 2 3 n +1 then from F (ai,ς)=G (ai,z1) follows |F (ai,ς)| = |G (ai,z1)|. As a little 2 while ago, we deﬁne limits max |F (a1,... ,an,ς)| = |a1||ς| + |a2||ς| + ···+ 2 n | || |n | | | | |z1| | | |z1| ··· | | |z1| an ς and max G (a1,... ,an,z1) = a1 2 + a2 3 + + an n+1 . From |F (ai,ς)| = |G (ai,z1)| ,i= 1,n, specially for |a1| = |a2| = ...= |an|, appears |z | |z | |z | |z | 1 √1 √1 √ 1 |ς| = = = 3 = ...= n , which is contradictory. So, none of above 2 3 4 n+1 mentioned equality is valid, but only |ς| < |z1|. 1824 J. Vujakovi´c

3 Sturm functions

Now we can solve integrals in 2 2 2 w1 =1− a (z) dz + a (z) dz a (z) dz ···+ L L L L L L n +(−1) a (z) dz2... a (z) dz2 (3.1) L L L L n-double integrals and 2 2 2 w2 = z − za(z) dz + a (z) dz za(z) dz −···+ L L L L L L n +(−1) a (z) dz2... za(z) dz2 (3.2) L L L L n-double integrals using theorem on mean value. We have the same situation as with iteration for real integrals, regardless of not having monotony (for details see [1],[3]) . z2 2 (2 1) 1 a (z) dz = a (z) dz dz = (a (ς1) z) dz = a (ς1) zdz = a (ς1) 2! L L L L L L 4 2 2 z1 a (z) dz a (z) dz = a (ς1) a (ς2) 4! L L L L 6 2 2 2 z1 a (z) dz a (z) dz a (z) dz = a (ς1) a (ς2) a (ς3) 6! L L L L L L . . z2n 2 ··· 2 1 a (z) dz a (z) dz =a (ς1) a (ς2) ...a(ςn) (2n)! L L L L n-double integrals where all ςi,i= 1,n are placed in the same circle,|ςi| < |z1|≤R1, but can have diﬀerent arguments. Besides that, when i → +∞ all ς i on their path tend to z1, staying in the circle of radius R1. Therefore, for (3.1) we obtain

2 4 6 z1 z1 z1 w = cosa z z =1− a (ς ) + a (ς ) a (ς ) − a (ς ) a (ς ) a (ς ) + ··· 1 ( ) 1 2! 1 2 4! 1 2 3 6! The mean value theorem... 1825

or ∞ k + 2k k z1 w = (−1) a (ςi) . (3.3) 1 (2k)! k=0 i=1

We notice that in coeﬃcients appear geometric mean in some mid points ςi, inside circle |z| < |z1| = R1. We shall mark these means in order with

g0 =1 g1 = a (ς1) g2 = a (ς 1) a (ς 2) 3 g3 = a (ς1) a (ς 2) a (ς 3) . . n gn = a (ς1) a (ς2) ...a(ς n) .

Even though the roots of complex number are multimorphic functions, they are in a certain way unambiguous, because all ςi,i= 1,n are inside the circle |z| < |z1| = R1, that is, if n → +∞ then series (3.3) gains more and → n → more members, so ς n z1. Here appears that a (ς 1) a (ς2)...a(ςn) n → n a (z1) a (z1) also for any branch of the function gn = a (z1). So n n lim a (ςk) = lim gn = a (z1) applies. We have now that n→+∞ n→+∞ k=1

∞ √ k 2 4 6 + 2 z1 2 z1 3 z1 k gkz1 w = cosa z z =1− g + g − g + ···= (−1) . 1 ( ) 1 2! 2 4! 3 6! (2k)! k=0 √ n Let’s also introduce geometric mean of these means g (ς)= g1g2 ...gn .In such way, upper sum becomes

2k +∞ g (ς)z k 1 w = cosa z z ≈ (−1) = cos g (ς)z , (3.4) 1 ( ) (2k)! 1 k=0 where symbol for approximately equal is valid because we have introduced one mean instead g1, g2, ... , gn. For solution w2 of equation w + a (z) w = 0, given with (3.2) with the same procedure we have similar estimations of multiple integrals 1826 J. Vujakovi´c

3 2 z1 za(z) dz = a (ς 1) 3! L L 5 2 2 z1 a (z) dz za(z) dz = a (ς1) a (ς2) 5! L L L L z7 2 2 2 1 a (z) dz a (z) dz za(z) dz = a (ς 1) a (ς 2) a (ς 3) 7! L L L L L L . . 2n+1 2 2 z1 ··· n a (z) dz za(z) dz = a (ς 1) a (ς 2) ...a(ς ) (2n+1)! . L L L L n-double integrals Here we also ﬁnd second sine solution

+∞ k 2k+1 k z1 w = sina z z = (−1) a (ς i) . (3.5) 2 ( ) (2k + 1)! k=0 i=1

If in the same way as before we introduce geometric means we obtain

2k+1 +∞ g (ς)z 1 k 1 1 w = sina z z ≈ (−1) = sin g (ς)z . 2 ( ) (2k + 1)! 1 g (ς) k=0 g (ς)

Deﬁnition. Functions w1 and w2 given with (3.3) and (3.5) we call Sturm complex function with the basis a(z), which is analytical function.

4 Conclusion

The diﬀerences between Sturm real (for the details see [1]) and complex (for the details see [4])functions are as follows: 1. Real functions are unambiguous while complex are not. 2. In real a (x) is positive, in complex a (z) is entire function. 3. In the case of real, solutions are whole; while in complex sinusoidal solution can have omitted points. 4. Solutions are limited when x →∞, and in complex do not have to be limited if z →∞. 5. For a (x) > 0 oscillatory solutions are strictly separated from monotonous, when a (x) < 0, while in the case of complex, due to absence of relations “less” and “greater” , such division does not exist. The mean value theorem... 1827

Contraction of operator for linear complex diﬀerential equation of second order w + a (z) w = 0, can easily be proven and the contraction coeﬃcient can easily be found. Analogy with analytic sine and cosine in real sense has been shown in this paper. The problem remained to be shown is the oscillation of the solution, that is, locating of zeros. Acknowledgments. Financial support for this study was granted by the Ministry of Education, Science and Technology Development of the Republic of Serbia (Project Number TR 35030).

References

[1] M. Leki´c, S. Cveji´c, D. Dimitrovski, Sturm theorems through iteration, Monography, University of Priˇstina, Faculty of Sciences and Mathematics, Kosovska Mitrovica, 2012. (in Serbian)

[2] B. V. Shabat, Introduction to complex analysis, Providence, R. I: American Mathematical Society, 1992.

[3] J. Vujakovi´c, M. Rajovi´c, D. Dimitrovski, Some new results on a linear equation of the second order, Computers and Mathematics with Applica- tions 61 (2011), 1837-1843, doi: 10.1016/j.camwa.2011.02.012.

[4] J. Vujakovi´c, Zeros solution of complex diﬀerential equations, Disserta- tion, University of Priˇstina, Faculty of Sciences and Mathematics, Kosovska Mitrovica, 2012. (in Serbian)

Received: February 5, 2014