A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^...A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^(2).Jacob Bernoulli first described the lemniscate in 1694.The Fagnano discovered the double angle formula of the lemniscate(1718).The Euler extended the Fagnano’s formula to a more general addition theorem(1751).The lemniscate function was subsequently proposed by Gauss around the year 1800.These insights were summarized by Jacobi as the theory of elliptic functions.A leaf function is an extended lemniscate function.Some formulas of leaf functions have been presented in previous papers;these included the addition theorem of this function and its application to nonlinear equations.In this paper,the geometrical properties of leaf functions at n=2 and the geometric relation between the angle θ and lemniscate arc length l are presented using the lemniscate curve.The relationship between the leaf functions sleaf_(2)(l)and cleaf_(2)(l)is derived using the geometrical properties of the lemniscate,similarity of triangles,and the Pythagorean theorem.In the literature,the relation equation for sleaf_(2)(l)and cleaf_(2)(l)(or the lemniscate functions,sl(l)and cl(l))has been derived analytically;however,it is not derived geometrically.展开更多
Here we discuss some phenomena of equiconvergence for the functions analytic inside the lemniscate. A quantitative estimate of sequences of differences between the Jacobi polynomials and Lagrange interpolants and some...Here we discuss some phenomena of equiconvergence for the functions analytic inside the lemniscate. A quantitative estimate of sequences of differences between the Jacobi polynomials and Lagrange interpolants and some other results are obtained.展开更多
According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n?1.Solving the ordinary differential equat...According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n?1.Solving the ordinary differential equations numerically results in waves appearing in the figures.The ordinary differential equation is very simple;however,waves,including the regular amplitude and period,are drawn in the figure.In this study,the function for obtaining the wave is called the leaf function.Based on the leaf function,the exact solutions for the undamped and unforced Duffing equations are presented.In the ordinary differential equation,in the positive region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes negative.Therefore,in the case that the curves vary with the time under the conditionχ(t)>0,the gradient dχ(t)/d constantly decreases as time increases.That is,the tangential vector on the curve of the graph(with the abscissa and the ordinate χ(t)changes from the upper right direction to the lower right direction as time increases.On the other hand,in the negative region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes positive.The gradient d χ(t)/d constantly increases as time decreases.That is,the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases.Since the behavior occurring in the positive region of the variable χ(t)and the behavior occurring in the negative region of the variableχ(t)alternately occur in regular intervals,waves appear by these interactions.In this paper,I present seven types of damped and divergence exact solutions by combining trigonometric functions,hyperbolic functions,hyperbolic leaf functions,leaf functions,and exponential functions.In each type,I show the derivation method and numerical examples,as well as describe the features of the waveform.展开更多
Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function ...Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.展开更多
In this paper we have generalized some results of Rahman [1] by considering the maximum of |f(z)| over a certain lemniscate instead of considering the maximum of|f(z)|, for |z|=r and obtain the analogous results for t...In this paper we have generalized some results of Rahman [1] by considering the maximum of |f(z)| over a certain lemniscate instead of considering the maximum of|f(z)|, for |z|=r and obtain the analogous results for the entire function |f(z)|=Σpk(z) [q(z)]k-1 where q(z) is a polynomial of degree m and pk(z)is of degree m-1. Moreover, we have obtained some inequalities on the lover order, type and lower type in terms of polynomial coefficients.展开更多
基金supported by Daido University research Grants(2020).
文摘A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^(2).Jacob Bernoulli first described the lemniscate in 1694.The Fagnano discovered the double angle formula of the lemniscate(1718).The Euler extended the Fagnano’s formula to a more general addition theorem(1751).The lemniscate function was subsequently proposed by Gauss around the year 1800.These insights were summarized by Jacobi as the theory of elliptic functions.A leaf function is an extended lemniscate function.Some formulas of leaf functions have been presented in previous papers;these included the addition theorem of this function and its application to nonlinear equations.In this paper,the geometrical properties of leaf functions at n=2 and the geometric relation between the angle θ and lemniscate arc length l are presented using the lemniscate curve.The relationship between the leaf functions sleaf_(2)(l)and cleaf_(2)(l)is derived using the geometrical properties of the lemniscate,similarity of triangles,and the Pythagorean theorem.In the literature,the relation equation for sleaf_(2)(l)and cleaf_(2)(l)(or the lemniscate functions,sl(l)and cl(l))has been derived analytically;however,it is not derived geometrically.
文摘Here we discuss some phenomena of equiconvergence for the functions analytic inside the lemniscate. A quantitative estimate of sequences of differences between the Jacobi polynomials and Lagrange interpolants and some other results are obtained.
文摘According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n?1.Solving the ordinary differential equations numerically results in waves appearing in the figures.The ordinary differential equation is very simple;however,waves,including the regular amplitude and period,are drawn in the figure.In this study,the function for obtaining the wave is called the leaf function.Based on the leaf function,the exact solutions for the undamped and unforced Duffing equations are presented.In the ordinary differential equation,in the positive region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes negative.Therefore,in the case that the curves vary with the time under the conditionχ(t)>0,the gradient dχ(t)/d constantly decreases as time increases.That is,the tangential vector on the curve of the graph(with the abscissa and the ordinate χ(t)changes from the upper right direction to the lower right direction as time increases.On the other hand,in the negative region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes positive.The gradient d χ(t)/d constantly increases as time decreases.That is,the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases.Since the behavior occurring in the positive region of the variable χ(t)and the behavior occurring in the negative region of the variableχ(t)alternately occur in regular intervals,waves appear by these interactions.In this paper,I present seven types of damped and divergence exact solutions by combining trigonometric functions,hyperbolic functions,hyperbolic leaf functions,leaf functions,and exponential functions.In each type,I show the derivation method and numerical examples,as well as describe the features of the waveform.
文摘Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.
文摘In this paper we have generalized some results of Rahman [1] by considering the maximum of |f(z)| over a certain lemniscate instead of considering the maximum of|f(z)|, for |z|=r and obtain the analogous results for the entire function |f(z)|=Σpk(z) [q(z)]k-1 where q(z) is a polynomial of degree m and pk(z)is of degree m-1. Moreover, we have obtained some inequalities on the lover order, type and lower type in terms of polynomial coefficients.
