Introducing Bernoulli Integral For Solving Some Physical Problems

Some of both modeling and problems in physics have been commonly presented in a first-order nonlinear differential equations (DE) of constant coefficients. Because the DE are integrable, therefore one must have an integral formulation for solving the physical problems. This paper introduces Bernoulli integral to complete the Tables of Integral for all of the Mathematical Handbooks.

Basically, the Bernoulli integral is integral form of the homogeneous Bernoulli differential equation (BDE) of constant coefficients. Under proper transformation, the Bernoulli integral can be used to generate another integral formulation especially for integrals that can be transformed into arctangent DE. By using the Bernoulli integral, one can create its self the integral formulation of solving the physical problems, and hence reduces utilization the tables of integral. A special application in generating Euler formula also presented.

Introduction

Some of both modeling and problems in physics have been commonly presented in a first-order nonlinear differential equations (DE) of constant coefficients For instance, in designing electromagnetic apparatus [Markus,1979], the logistic growth process [Welner,2004], chaotic behavior [Barger et al,1995], the generation and propagation of soliton [Wu et al 2005],[Morales,2005], the transport of fluxon [Gonzile et al,2006], the generation of squeezed laser [Friberg,1996 ],etc. One requires Table of Integral to solve a specific integral for solving such differential equation [Spiegel,MR,1968]. To complete the Table of integral, we introduce Bernoulli integral that until now not including in both of the Table integral and mathematical Handbook. By using the Bernoulli integral, one can create the integral formulation required in solving the physical problems, and hence reduces utilization the Tables of integral.

Key-words : Arctangent, tangent, arctangent differential equation, Bernoulli equation, Bernoulli differential equation, integral, Bernoulli integral, Schrödinger equation, modulation instability, Euler formula, Argand diagram, electromagnetic, logistic growth, chaotic, soliton, fluxon, squeezed laser

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Fighting of the Cause of Allah by Governing a Smart Mathematics Based on Islamic Teology

Existence of the universe is reality evidence of supremacy and science fame of God Allah SWT. Inking seven of times water in all ocean world (even more) not enough to write down it. According to writer, mathematical model which representatively as stepping in developing the Islamic Scientific is arctangent differential equation

Eqs.(1)

which its exact solution is of the form:

Eqs.(2)

Because of writer looks into this arctangent differential equation is having the religion character
(according to writer that for a=1, b=1, and both of initial values t₀ = 0 dan y₀ = 0 the value of the tangent function at correspond to the Qidam and Baqa properties), hence solution yielded a solver technique entering religion factors must still appropriate to the exact solution

This paper introduces a new technique of solving a nonlinear first order ordinary differential equation so-called as SMT (stands for Stable Modulation Technique) which its solution is in the form of AF(A), that is a formula of modulation function which its amplitude term is also including in the phase function. The transfromation function applied for solving eq.(1) by using SMT is what gives its final solution in the form :

Eqs.(3)

The idea of developing this stable modulation technique based on the event of Isra' and Mi’raj of prophet Muhammad, which alongside its journey towards Sidhratulmuntaha guided by angel Jibril. Eqs.(3) assures writer that when mi’raj the energy of prophet Muhammad is transferred into the energy form of modulated wave. The fundamental aspect for developing of modern mathematics and computing is obtained when to = 0, yo = 0, a = 0 dan b = 0 where eq.(3) then reduces to the form :

Eqs .(4)

Eqs.(4) as a representative form of tangent function up to now has not been met in Mathematics Handbook, because the only

Eqs .(5)

But both of eq.(4) and eq.(5) are still giving the same value with the value of tan(t) for all values of t except at t = pi / 2 in eq.(4) and at t = pi in eq.(5) which both giving value of 0/0, though value of tan(pi/2) = ~. In mathematics the value of 0/0 is unknown as commonly called as NaN (stands for Not a Number). The value of ~ is still not obtaining from eq.(4) and eq.(5), even if has been performed the limit operation because it is only giving devide by zero:

Eqs .(6)

At presentation of the exact solution of arctangent differential equation brightens the confidence of writer that during journey Isra', angel Jibril telling the exact properties of God, while during journey Mi’raj of prophet Muhammad is supplied by a stabilization of believe that God doesn't spell out members as apparently at 0/0, and man will never can reach God will desire, as apparently at 1/0. The primary message is that mathematics applied as "approach" properly in the effort of explaining the Sunnatullah, and don't make mathematics as a justification tool.

Keywords : Jihad, Isra’ Mi’raj, Prophet Muhammad, Angel Jibril, mathematics, arctangent, arctangent differential equation, tangent function, NaN (0/0), devide by zero (1/0), sunnatullah

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Anti Einstein Technique : Analytic Solution of Nonlinear Schrödinger Equation by Means of A New Approach

The nonlinear Schrödinger equation (NSE) has served as the governing equation of optical soliton in the study of its applications to optical communication and optical switching. Various schemes have been employed for the solution of this nonlinear equation as well as its variants. We report in this paper a relatively simpler new approach for the analytic solution of NSE. In this scheme the equation was first transformed into an arctangent differential equation, which was then separated into the linear and nonlinear parts, with the linear part solved in a straight forward manner. The solution of the nonlinear equation was written in the form of modulation function characterized by its amplitude function A and phase function F(A). Substituting the linear solution for A, the arctangent differential equation was solved for a certain initial value of A. It is shown that this method is applicable to other first-order nonlinear differential equation such as the Korteweg de Vries equation (KdV), which can be transformed into an arctangent differential equation.

I. Introduction

The phenomenon of the solitary wave propagation was observed for the first time by the Scottish scientist John Scott Russell in 1844, when one day he was watching water waves of a certain shape kept on traveling without changing their shape for a distance as far as his eye could see. To explain the behavior of such unusual wave, Korteweg and de Vries governed a model for the wave propagation in shallow water in form a partial differential equation called as KdV differential equation, which its solution appropriates to the features of the solitary wave called as soliton[1]. The existence of solitons in optical fiber was predicted by Zakarov and Zabat (1972) after they derived a differential equation for the light propagating in an optical fiber, that demonstrated later by Hazagawa in 1973 at Bell Laboratory. Next, Mollenauer and Stolen employed the solitons in optical fiber for generating subpicosecond pulses.

Keywords : nonlinear Schrödinger equation, KdV, arctangent, Anti Einstein Technique

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Introducing Stable Modulation Technique for Solving an Inhomogeneous Bernoulli Differential Equation

The Bernoulli differential equation that has been used as primary modeling in many application branches commonly contains harmonic force function in the inhomogeneous term. The nhomogeneous Bernoulli differential equation (IBDE) is specified by nonlinearity of nth order. For instance, 3rd order IBDE is called a stochastic differential equation that has been commonly applied for describing the corrosion mechanism, the transport of fluxon, the generation of squeezed laser, etc. Due to the difficulty of applying a linearization procedure for IBDE, it has commonly been solved numerically. In this paper we introduced a so-called Stable Modulation Technique (SMT) which is able to solve a first order nonlinear differential equation that transformable into the homogeneous Bernoulli differential equation (BDE). SMT is employed by splitting BDE into linear and nonlinear parts. The solution of its nonlinear part has been found to be AF(A) where A is the initial value of nonlinear solution part and F is the modulation function whose phase is a function of A. The general solution of BDE obtained by substituting the linear solution part into initial value of the nonlinear solution part. Although IBDE can not be transformed into BDE completely, SMT gives nevertheless an approximation solution in AF(A) form, where the homogeneous solution part becomes its amplitude term. The AF(A) formula for stochastic differential equation with cosine function as inhomogeneous term can be decomposed as well into transient and steady state solutions. In addition, a special example of solving 2 ogeneous Bernoulli differential equation (BDE) has en used as a primary modeling scheme in many ation branches. A nonlinear BDE is specified by a nonlinearity of n. Although the stochastic differential equation is the first order differential equation, nevertheless the procedure of obtaining its analytical solution is very complicated as shown in utilizing of BWK (Brillouin, Wenzel, Kramer) and Reversion methods nd order BDE in creating a new Planck’s formula of black-body radiation is also presented.

Keywords–Stable modulation technique, modulation function, stochastic differential equation.

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Analytical Solution Of The Ricatti Differential Equation For High Frequency Derived By Using The Stable Modulation Technique

The Ricatti differential equation dy/dt = P(t)y2+Q(t)y+R(t) is a nonlinear differential equation which is of contemporary interest in various fields, including particle dynamics, optics, and petroleum exploration. The Ricatti differential equation is easily solved by numerical methods. But in order to obtain the exact solution in analytical form, the first order of nonlinear inhomogeneous differential equation is commonly convert into a second order linear differential equation by use of a change of the dependent variable. In this paper we introduce the stable modulation technique (SMT) to solve the Ricatti differential equation without of the use linearization procedure. The main principle of the SMT in solving a first order nonlinear differential equation is modulate the solution of the linear part into the initial value of the nonlinear part solution. Important to be stressed here that the solution of nonlinear part must be written in the modulation function, where the initial value acts as amplitude and also including in the total phase shift. For a special case, dy/dt = -by2+ay+Acos(2πft) where a and b of both are constants, while t is variable of the time (in s), frequency f in Herzt (Hz) we find that the analytical solution of the SMT is appropriate with numerical solution espescially for high frequency f≥10 Hz, amplitude value A≤1, and initial value of the y in the range 0.1≤y0≤1. The analytical solution above can be used as trial function when the Ricatti differential equation will be solved by using combination of the modulational instability technique and variational approximation.

Keywords : Nonlinear inhomogeneous differential equation, linearization procedure, stable modulation technique, modulational instability

I. Introduction

The general form of the Ricatti differential equation (DE) is of the form :
dy/dt = P(t)y2+Q(t)y+R(t), .......................................(1)
where y and t are respectively dependent and independent variables, both of P(t) and Q(t) are homogeneous coefficients, where R(t) is the inhomogeneous term[1]. The Ricatti DE is mother of all ordinary differential equations (ODE’s) second order generating special functions like Airy, Bessel and etc,[2] even the Helmholtz equation[3] which has broad applications including optics and geophysics. Common procedure in solving the Ricatti DE is by transforming into a linear ODE second order[1],[2],[4]. For special case, the analytical exact solution of the Ricatti DE can not be obtained, because the analytical solution of the corresponding linear ODE second order is not available.

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How can prove that zero equal zero (0=0) or (1=1)

Who want to know and solve this equation?
if anybody can solve this problem and explain this equation, maybe you the best on matematic

1 = 1
a = a
(a**2 - a**2) = (a**2 - a**2)

..........................................
..........................................
..........................................
the result must be 1 = 1 ,but
the result is 1 = 2

you can answer on comment at the sidebar

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Analytical Formulation of Normal Modes in Symmetrical Directional-coupler

The analysis of the optical power transfer in the linear step index directional-coupler based on the couple-mode theory is inaccurate for a small gap. This problem has been previously overcome by using the normal-modes approximation. Commonly,

this approximation has been solved by numerical methods such as Fourier transform or finite difference. In this paper, the Helmholtz equation is, instead, analytically solved by using a characteristic matrix of multiplayer waveguides in order to find the electric field and its propagation constant of the normal-modes. The importance of these analytical formulas, is that a phase shift of the normal modes along the propagation can be easily analyzed.

1. Introduction

In integrated optics areas, the directional-couplers are the major interest with potential applications to optical communications, i.e, used to fabricate low-loss optical switches[1], high speed modulators[2], polarization splitter[3] and wavelength demultiplexer/multiplexer[4]. Due to coupling effect, optical power can be transferred from one waveguide to another adjacent waveguide as a result of the overlap in the evanescent fields of the two guides. The amount of power transferred between the waveguides depends upon the waveguide parameters, i.e, the guided wavelength, the confinement of the individual waveguides, the separation between them, the length over which they interact, and the phase mismatch between the individual waveguides[5].

The power transfer of two waveguides in the directional-couplers has been treated extensively utilizing the coupled-mode method, but as shown in [6] this method becomes less accurate when the waveguides get too close. An alternative choice is the normal-modes approximation. This approximation taken full account of the entire structure and solves for modal indices and guided fields of the supermodes. In the normal-modes approach, the characteristic of the directional-couplers are then represented by interferences between the guided fields of the supermodes[7], i.e symmetrical and asymmetrical modes. In practical, the directional-couplers are made in 3-D structure, consist of waveguides with finite lateral dimensions. In order to obtain the exact solutions of normal modes, the 3-D is usually reduced to 2-D guides structure[7],[8]. Hence in the 2-D guides, the two parallel waveguides with their surrounding medium can be considered as a single structure, so that the normal-modes of the such structure can be solved by method of multilayer waveguides. In this paper we use the multilayer waveguides to formulate the optical electric fields in the symmetrical directional-couplers. The expression of such the guided fields derived by method of multilayer waveguides given by Kogelnik[9], and Rohedi[10].

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Application of Stable Modulation Scheme for Solving Bernoulli Differential Equation

Solving of Bernoulli differential equation traditionally always is done applies linearization procedure by using Bernoulli transformation function. This paper introduces a new technique of solving the Bernoulli differential equation without using linearization by application of stable modulation scheme.

Application of the method named Stable Modulation Technique (called as SMT) is started by splitting the Bernoulli differential equation to parts of linear and nonlinear, then writes down the solution of nonlinear part in the form of modulation function which its initial value besides played the part of as amplitude A and also is modulated into a phase function F(A). The exact solution of Bernoulli differential equation given in AF(A) formula obtained after replacing the linear solution part into initial value of its nonlinear part solution. In this paper presented the usage of SMT for solving the storage model of magnetic energy into inductor.

I. Introduction
The homogeneous Bernoulli differential equation that commonly called Bernoulli differential equation (BDE) to become as primary model in so many application branches. The BDE is distinguished to the degree of its nonlinearity (n). For instance, the BDE having degree of two commonly applied to model growth of logistic in Biology[1] and the behavior of chaos[2], while for the degree of three (n=3)
the BDE forms Gizbun or quartic equation commonly used to analyze corrosion process[3]. The BDEalso is nonlinear part of Klein Gordon partial differential equation which is the usage widely, among these are in studying the dynamics of elementary particles and stochastic resonances4], the transportation of fluxon[5], the excitation of squeezed laser[6], etc.

As commonly explained in mathematical handbook[7],[8], solving of BDE always is done through linearization procedure as in recommending by Jacob Bernoulli. The transformation from the form of nonlinear to the linear differential equation is performed by using Bernoulli transformation function, and hereinafter solved by using the common method of solving a linear differential equation. Recently, Rohedi[9] has reported verification of the Bernoulli transfomation function, and justify the general solution of Bernoulli differential equation which written in mathematical handbook. At the paper was introduced stable modulation technique (called as SMT) focused to solve BDE of constant coefficients, especially which its solution is started from ordinary point. Rohedi[10] has also reported application of SMT for solving a Ricatti differential equation of constant coefficients which its inhomogeneous term in form of sinusoidal function that also was started from ordinary point. In this paper, applying the SMT is developed to solve BDE for arbitrary value of its linear and nonlinear coefficients, either and also constant valuable and varying as function of its dependent variable. In mathematics, this differential equation is known as the general homogeneous Bernoulli differential
equation.

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Simplified Calculation of Guided Nonlinear Boundary-Wave Parameters Using Optimization Procedure

A guided wave excited along the boundary between linear and nonlinear media known as initial inspiration for developing devices based on Kerr nonlinear optics, such as the nonlinear directional coupler, etc. Two important parameters for such structure are respectively the minimum amplitude of light required for the excitation, and the location of the peak of guided nonlinear boundary wave. Analytical procedure of derivation the two parameters commonly involved the Jacobi’s elliptic functions based on the numerical integration. In order to simplify the calculation procedure, in this paper we introduce optimization procedure based on applying the solitary wave solution for guided field inside the nonlinear media. The simulation of guided wave excitation at the interface between linear and nonlinear media is also presented

1. Introduction

In integrated optics, all of optical devices have been made in waveguide structures, based on both linear and nonlinear optics materials. The simpliest structure of optical waveguides made of linear materials whose all of refractive indices independently to the propagating light intensity requires three layers that known as slab optical guide or slab waveguide [1]. Hence, an advanced devices such as directional-coupler that commonly applicable for optical power transfer and/or optical switching, beside in complicated structure, but also it requires applying of external treatments, because all devices made of linear materials only can operate as passive components [2]. On the other hand, due to the dependency to the propagating light intensity, recently the nonlinear materials especially for the Kerr optics materials much be applied for fabricating active-optical waveguides, that have been commonly used in many application branches, for examples, all devices of X-junction, Mach-zender interferometer, feedback grating, optical bistability, etc [3]. In addition, because of self focusing of the nonlinear Kerr optics materials, the number layer of slab waveguide reduces from three into two layers, while optical wave that propagates over the waveguide is commonly called as the boundary wave. Important to be stressed here, that the two-layer slab waveguide consists of a nonlinear Kerr optics material as the guiding layer is deposited on top a substrate of linear optics material [4].
The main problem in designing the two layers slab waveguide that are determination of the minimum amplitude of light required for the excitation, and the location of the peak of guided nonlinear boundary wave. The two parameters are depend on the effective refractive index of the slab waveguide. This paper introduces optimization procedure for obtaining all parameters of the two layer slab waveguide. The procedure of optimization was primarly applied to maximize the peak of electric field guided boundary wave in the nonlinear guiding layer.

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Creating A New Planck’s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram

This paper reports derivation of a new Planck’s formula of spectral density of black-body radiation, that was originated by modeling the interpolation formula of Planck’s law of obtaining the mean of energy of black-body cavity in 2^nd order of Bernoulli equation. The new Planck’s formula is created by means AF(A) diagram of solving arctangent differential equation after transformin the Bernoulli equation into the arctangent differential equation The New Planck’s formula not only contains the terms of the photon energy and the energy difference between two states of the motion of harmonic oscillator (), but also contains both terms of the minimum energy of harmonics oscillator () and the phase differences () as representing the intermodes-orthogonality, hence it can answer why the explanation of black-body radiation has been associated with the harmonic oscillators.

I. Introduction

The era of developing the modern scientific was originated by presence the Planck’s law of black-body radiation as representation of the light sources in a thermal equilibrium. Planck not only could complete the Rayleigh-Jeans and Wien’s laws, both of radiation laws previously that of each only appropriates to the experimental for the range of long wavelength and short wavelength respectively, but he also created a new constant h called as Planck’s constant that not known previously in classical physics [1]. Based on his constant, Planck postulated the discretitation of electromagnetic energy in packet of energy called as photon, where for every angular frequency (), the energy per photon is , which was justified later by Einstein through fotoelectric effect [2]. Planck’s law for black-body radiation also became the primary base of derivation Einstein coefficients of spontaneous and stimulated of emission rates for generating the light sources such as maser and laser [3].

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How to upgrade the running time of Computer

One of built-in function required in building micro-processor of computer is infinite series of tangent function. Until now, there is one general formula for the infinite series of tangent function that available in mathematical handbook, and also used in all symbolic software-package. The general formula was created by Sir. Bernoulli that is of form

where,is Bernoulli numbers, and n 1,2,3,. Unfortunately, the general formula is not consistent with Maclaurin series that always contains n ! in denominator of each terms. In this paper, we present Rohedi’s reversion for obtaining the infinite series of tangent function without of use the Maclaurin series, but its result is still consistent with the Maclaurin series. Derivation of RohediSmart reversion formula based on solution of the arctangent differential equation

solved ecursively by using short stable modulation technique (S-SMT). Comparison the infinite series of tangent function of Rohedi’s reversion with the result of both Matematica 5.1, Maple 9.5 shows thattime consuming of Rohedi’s reversion is shortest, hence need smallest of computational memory.Finally, we give comparison the time consuming of Rohedi’s reversion for 4.0365 10321 x1635of 1635nd coefficient of infinite series of tangent function calculated using matlab needs2.744 s, while Maple 9.5 soft needs 36.35 s

Discussion and Conclusion

We show that output of Rohedi’s reversion for infinite series of tangent function is still consistent with Maclaurin series. According to the equality of output both of Matematica and Maple software-package, we resume that both software-package have been used equal general formula, that is Bernoulli’s formula. Hence, we take a conclusion that Rohedi’s reversion formula as a new general formula of infinite series of the tangent function that can be used to upgrade the running time of computer.

References :

1. Abranowitz,M., and Stegun, I.A., “Handbook of Mathematical Functions”, New-York, 1972
2. Spiegel,M.R.,”Mathematical Handbook of Formulas and Tables”, Schaum’s Outline Series,McGRAW-Hill Book Company, page 104, 1968.
3. Rohedi,A.Y., “Solving of the homogeneous nonlinear differential equation by using Stable Modulation Technique”, Presented on Conference of Mathematical Analysis and its Applications”, Department of Mathematics, Natural Sciences, ITS, Surabaya, Indonesia, 10-11 August 2006.
4. Rohedi,A.Y., “Analytical Solution Of The Ricatti Differential Equation For High Frequency Derived By Using The Stable Modulation Technique”, Presented on International Conference of Mathematics and Natural Sciences, Poster Edition, Faculty of Natural Sciences, ITB, Bandung, Indonesia, 29-30 Nopember 2006.
5. Shortcut Solution for Bernoulli Equation in AF(A) Formula Based on Stable Modulation Technique (will be submitted for publication

This paper will be submitted for publication.

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Letter from Rohedi Laboratory (Laboratory of New Science)

Dear All,

I interest to investigate the validity of output symbolic software especially for calculating the roots of polynomials . Now, I have been developing a new method for the purpose. The method can create analytical formulation for n order polynomials,

To show the performance of the method, firstly, I present the comparison the first root of the following polynomial to the matlab result.

From ROHEDI Laboratory Surabaya

>>polinomn([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,

28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,500,51,52,53,

54,55,56,57,58,59,600,61,62,63,64,65,66,67,68,69,700,71,72,73,74,75,76,77,78,79,

800,81,82,83,84,85,86,87,88,89,900,91,92,93,94,95,96,97,98,99,1000,101,102,103,

104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,

124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,

144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,

164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,

184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,

204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,

224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,

244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,

264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,

284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,

304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,

324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,

344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,

364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,

384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,

404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,

424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,

444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,

464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,

484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,

504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,

524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,

544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,

564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,

584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,

604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,

624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,

644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,

664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,

684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,

704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,

724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,

744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,

764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,

784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,

804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,

824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,

844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,

864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,

884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,

904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,

924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,

944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,

964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,

984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,

1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,

1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,

1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,

1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061])

The first root of 1060 order polynomial above is

The root of matlab = 0.90967612057974 + 0.65233483663790i

with remainder = -5.386242315198940e+040 +3.112117014950832e+040i

matlab time consuming = 28.711 s.

myroot= -0.99461160642852 + 0.10321236597345i

with remainder = 9.436007530894131e-010 -5.946915848653589e-009i

my time consuming = 3.725 s.

My formula not only very accurate but also requires least of time consuming.

Here, I need your information what is the appropriate journal for publication the method. Or, if any one or company who interest to buy the formula, you can contact me via email :afasmt@yahoo.com

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