Euler's Identity Research Articles

Complex-Type Numbers and Generalizations of the Euler Identity

We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural way in algebraic and geometric terms.

Number theory: a lively introduction with proofs, applications, and stories

Preface. To the Student. To the Instructor. Acknowledgements. 0. Prologue. 1. Numbers, Rational and Irrational. (Historical figures: Pythagoras and Hypatia). 1.1 Numbers and the Greeks. 1.2 Numbers you know. 1.3 A First Look at Proofs. 1.4 Irrationality of he square root of 2. 1.5 Using Quantifiers. 2. Mathematical Induction. (Historical figure: Noether). 2.1.The Principle of Mathematical Induction. 2.2 Strong Induction and the Well Ordering Principle. 2.3 The Fibonacci Sequence and the Golden Ratio. 2.4 The Legend of the Golden Ratio. 3. Divisibility and Primes. (Historical figure: Eratosthenes). 3.1 Basic Properties of Divisibility. 3.2 Prime and Composite Numbers. 3.3 Patterns in the Primes. 3.4 Common Divisors and Common Multiples. 3.5 The Division Theorem. 3.6 Applications of gcd and lcm. 4.The Euclidean Algorithm. (Historical figure: Euclid). 4.1 The Euclidean Algorithm. 4.2 Finding the Greatest Common Divisor. 4.3 A Greeker Argument that the square root of 2 isIrrational. 5. Linear Diophantine Equations. (Historical figure: Diophantus). 5.1 The Equation aX + bY = 1. 5.2 Using the Euclidean Algorithm to Find a Solution. 5.3 The Diophantine Equation aX + bY = n. 5.4 Finding All Solutions to a Linear Diophantine Equation. 6. The Fundamental Theorem of Arithmetic. (Historical figure: Mersenne). 6.1 The Fundamental Theorem. 6.2 Consequences of the Fundamental Theorem. 7. Modular Arithmetic. (Historical figure: Gauss). 7.1 Congruence modulo n. 7.2 Arithmetic with Congruences. 7.3 Check Digit Schemes. 7.4 The Chinese Remainder Theorem. 7.5 The Gregorian Calendar. 7.6 The Mayan Calendar. 8. Modular Number Systems. (Historical figure: Turing). 8.1 The Number System Zn: an InformalView. 8.2 The Number System Zn: Definitionand Basic Properties. 8.3 Multiplicative Inverses in Zn. 8.4 Elementary Cryptography. 8.5 Encryption Using Modular Multiplication. 9. Exponents Modulo n. (Historical figure: Fermat). 9.1 Fermat's Little Theorem. 9.2 Reduced Residues and the Euler \phi-function. 9.3 Euler's Theorem. 9.4 Exponentiation Ciphers with a Prime modulus. 9.5 The RSA Encryption Algorithm. 10. Primitive Roots. (Historical figure: Lagrange). 10.1 Zn. 10.2 Solving Polynomial Equations inZn. 10.3 Primitive Roots. 10.4 Applications of Primitive Roots. 11. Quadratic Residues. (Historical figure: Eisenstein) 11.1 Squares Modulo n 11.2 Euler's Identity and the Quadratic Character of -1 11.3 The Law of Quadratic Reciprocity 11.4 Gauss's Lemma 11.5 Quadratic Residues and Lattice Points. 11.6 The Proof of Quadratic Reciprocity. 12. Primality Testing. (Historical figure: Erdos). 12.1 Primality testing. 12.2 Continued Consideration of Charmichael Numbers. 12.3 The Miller-Rabin Primality test. 12.4 Two Special Polynomial Equations inZp. 12.5 Proof that Millar-Rabin is Effective. 12.6 Prime Certificates. 12.7 The AKS Deterministic Primality Test. 13. Gaussian Integers. (Historical figure: Euler). 13.1 Definition of Gaussian Integers 13.2 Divisibility and Primes in Z[i]. 13.3 The Division Theorem for the Gaussian Integers. 13.4 Unique Factorization in Z[i]. 13.5 Gaussian Primes. 13.6 Fermat's Two Squares Theorem. 14. Continued Fractions. (Historical figure: Ramanujan). 14.1 Expressing Rational Numbers as Continued Fractions. 14.2 Expressing Irrational Numbers as Continued Fractions. 14.3 Approximating Irrational Numbers Using ContinuedFractions. 14.4 Proving that Convergents are Fantastic Approximations. 15. Some Nonlinear Diophantine Equations. (Historical figure: Germain). 15.1 Pell's Equation 15.2 Fermat's Last Theorem 15.3 Proof of Fermat's Last Theorem for n = 4. 15.4 Germain's Contributions to Fermat's Last Theorem 15.5 A Geometric look at the Equation x4 +y4 = z2. Appendix: Axioms of Number Theory. A.1 What is a Number System? A.2 Order Properties of the Integers. A.3 Building Results From Our Axioms. A.4 The Principle of Mathematical Induction.

Frequency variant of Euler type identities and the problem of sign-constancy of the kernel in associated quadrature formulas

In the recent years many authors used extended Euler identities to obtain generalizations of some classical quadrature formulas with the best possible error estimates. The main step in obtaining the best possible error estimates was to control zeros of the kernel in the error term which consists of the affine combinations of the transates of periodic Bernoulli polynomials. The main goal of this paper is to consider a general case. The frequency variant of extended Euler identities is found more tractable for this problem.

On families of quadrature formulas based on Euler identities

A family consisting of quadrature formulas which are exact for all polynomials of order ⩽5 is studied. Changing the coefficients, a second family of quadrature formulas, with the degree of exactness higher than that of the formulas from the first family, is produced. These formulas contain values of the first derivative at the end points of the interval and are sometimes called “corrected”.

Subdifferential representation of homogeneous functions and extension of smoothness in Banach spaces

In this paper, we mainly consider proximal subdifferentials of lower semicontinuous functions defined on real Hilbert space and Clarke’s subdifferentials of locally Lipschitzian functions defined on Banach space respectively, and obtain the generalized Euler identity of homogenous functions. Then, by introducing a multifunction Γ, we extend the smoothness of sphere and differentiability of norm function in Banach space.

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman–Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers–Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers–Szegö polynomials. We give a new formula for the homogeneous Rogers–Szegö polynomials h n ( x , y | q ) . We introduce a q-difference operator θ xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659–668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s n ( x , y ; b | q ) and derive their generating function by using the new homogeneous q-shift operator L ( b θ xy ) .

Essential math skills for engineers

Preface. 1 What Do Engineers Do? 2 Miscellaneous Math Skills. 2.1 Equations of Lines, Planes, and Circles. 2.2 Areas and Volumes of Common Shapes. 2.3 Roots of a Quadratic Equation. 2.4 Logarithms. 2.5 Reduction of Fractions and Lowest Common Denominators. 2.6 Long Division. 2.7 Trigonometry. 2.7.1 The Common Trigonometric Functions: Sine, Cosine, and Tangent. 2.7.2 Areas of Triangles. 2.7.3 The Hyperbolic Trigonometric Functions: Sinh, Cosh, and Tanh. 2.8 Complex Numbers and Algebra, and Euler's Identity. 2.8.1 Solution of Differential Equations Having Sinusoidal Forcing Functions. 2.9 Common Derivatives and Their Interpretation. 2.10 Common Integrals and Their Interpretation. 2.11 Numerical Integration. 3 Solution of Simultaneous, Linear, Algebraic Equations. 3.1 How to Identify Simultaneous, Linear, Algebraic Equations. 3.2 The Meaning of a Solution. 3.3 Cramer's Rule and Symbolic Equations. 3.4 Gauss Elimination. 3.5 Matrix Algebra. 4 Solution of Linear, Constant-Coeffi cient, Ordinary Differential Equations. 4.1 How to Identify Linear, Constant-Coeffi cient, Ordinary Differential Equations. 4.2 Where They Arise: The Meaning of a Solution. 4.3 Solution of First-Order Equations. 4.3.1 The Homogeneous Solution. 4.3.2 The Forced Solution for Nice f ( t ). 4.3.3 The Total Solution. 4.3.4 A Special Case. 4.4 Solution of Second-Order Equations. 4.4.1 The Homogeneous Solution. 4.4.2 The Forced Solution for Nice f ( t ). 4.4.3 The Total Solution. 4.4.4 A Special Case. 4.5 Stability of the Solution. 4.6 Solution of Simultaneous Sets of Ordinary Differential Equations with the Differential Operator. 4.6.1 Using the Differential Operator to Verify Solutions. 4.7 Numerical (Computer) Solutions. 5 Solution of Linear, Constant-Coeffi cient, Difference Equations. 5.1 Where Difference Equations Arise. 5.2 How to Identify Linear, Constant-Coeffi cient Difference Equations. 5.3 Solution of First-Order Equations. 5.3.1 The Homogeneous Solution. 5.3.2 The Forced Solution for Nice f ( n ). 5.3.3 The Total Solution. 5.3.4 A Special Case. 5.4 Solution of Second-Order Equations. 5.4.1 The Homogeneous Solution. 5.4.2 The Forced Solution for Nice f ( n ). 5.4.3 The Total Solution. 5.4.4 A Special Case. 5.5 Stability of the Solution. 5.6 Solution of Simultaneous Sets of Difference Equations with the Difference Operator. 5.6.1 Using the Difference Operator to Verify Solutions. 6 Solution of Linear, Constant-Coeffi cient, Partial Differential Equations. 6.1 Common Engineering Partial Differential Equations. 6.2 The Linear, Constant-Coeffi cient, Partial Differential Equation. 6.3 The Method of Separation of Variables. 6.4 Boundary Conditions and Initial Conditions. 6.5 Numerical (Computer) Solutions via Finite Differences: Conversion to Difference Equations. 7 The Fourier Series and Fourier Transform. 7.1 Periodic Functions. 7.2 The Fourier Series. 7.3 The Fourier Transform. 8 The Laplace Transform. 8.1 Transforms of Important Functions. 8.2 Useful Transform Properties. 8.3 Transforming Differential Equations. 8.4 Obtaining the Inverse Transform Using Partial Fraction Expansions. 9 Mathematics of Vectors. 9.1 Vectors and Coordinate Systems. 9.2 The Line Integral. 9.3 The Surface Integral. 9.4 Divergence. 9.4.1 The Divergence Theorem. 9.5 Curl. 9.5.1 Stokes' Theorem. 9.6 The Gradient of a Scalar Field. Index.

Alder's conjecture

In 1956, Alder conjectured that the number of partitions of n into parts differing by at least d is greater than or equal to that of partitions of n into parts ≡ ±1 (mod d + 3). The Euler identity, the first Rogers-Ramanujan identity, and a theorem of Schur show that the conjecture is true for d = 1, 2, 3, respectively. In 1971, Andrews proved that the conjecture holds for d = 2r – 1, r ≧ 4. In this paper, we prove the conjecture for all d ≧ 32 and d = 7.

Generalized Euler identity for subdifferentials of homogeneous functions and applications

In this paper, we mainly consider subdifferentials and basic subdifferentials of homogeneous functions defined on real Banach space and Asplund space respectively, and obtain the generalized Euler identity. As applications, we consider constrained optimization problems and several geometric properties of Banach space.

Quadrature formulae of Gauss type based on Euler identities

The aim of this paper is to derive quadrature formulae of Gauss type based on Euler identities. First, we derive quadrature formulae where the integral over [0,1] is approximated by values of the function in three points: x , 1 / 2 and 1 − x . As special cases, the Gauss 2-point formula, Simpson’s formula, dual Simpson’s formula and Maclaurin’s formula are obtained. Next, corrected Gauss 2-point formulae are derived and finally, the Gauss 3-point formulae and the corrected Gauss 3-point formulae are considered. We call “corrected” such quadrature formulae where the integral is approximated both with the values of the integrand in certain points and the values of its first derivative in the end points of the interval. Corrected formulae have a degree of exactness higher than the adjoint original formulae.

A Short Proof of Euler's Identity for Continuants

Km+n(x1 , . . . , xm+n)Kl(xm+1 , . . . , xm+l)−Km+l(x1 , . . . , xm+l)Kn(xm+1 , . . . , xm+n) = (−1)Km−1(x1 , . . . , xm−1)Kn−l−1(xm+l+2 , . . . , xm+n), m ≥ 1, l ≥ 0, n ≥ l + 1. (1) It is usually verified by induction on two out of the three parameters l , m, n . The proof given in the present paper is based on the fact that the determinant of a skew-symmetric 4×4 matrix (just as of any matrix of even order) is the square of some polynomial (the Pfaffian of the matrix):

On weighted Euler harmonic identities with applications

Aweighted Euler identities involving harmonic sequences of functions are established. Consequently various generalizations of Ostrowski inequality involving weighted integrals are obtained.

Expansion of Weighted Pseudoinverse Matrices in Matrix Power Products

On the basis of the Euler identity, we obtain expansions for weighted pseudoinverse matrices with positive-definite weights in infinite matrix power products of two types: with positive and negative exponents. We obtain estimates for the closeness of weighted pseudoinverse matrices and matrices obtained on the basis of a fixed number of factors of matrix power products and terms of matrix power series. We compare the rates of convergence of expansions of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudoinverse matrices. We consider problems of construction and comparison of iterative processes of computation of weighted pseudoinverse matrices on the basis of the obtained expansions of these matrices.

CAUCHY AUGMENTATION FOR BASIC HYPERGEOMETRIC SERIES

The authors present a technique of deriving basic hypergeometric identities from specializations using fewer parameters, by using the classical Cauchy identity on the expansion of the power of x in terms of the q-binomial coefficients. This method is referred to as ‘Cauchy augmentation’. Despite its simple appearance, the Cauchy identity plays a key role in parameter augmentation. For example, one can reach the q-Gauss summation formula from the Euler identity by using the Cauchy augmentation twice. This idea also applies to Jackson's 2φ1 to 3φ1 transformation formula. Moreover, a transformation formula analogous to Jackson's formula is obtained. 2000 Mathematics Subject Classification 05A30, 33D15.

An analogue of Euler's identity and new combinatorial properties of n-colour compositions

An analogue of Euler's partition identity: “The number of partitions of a positive integer ν into odd parts equals the number of its partitions into distinct parts” is obtained for ordered partitions. The ideas developed are then used in obtaining several new combinatorial properties of the n-colour compositions introduced recently by the author.

On generalizations of Ostrowski inequality via Euler harmonic identities

Some generalizations of Ostrowski inequality are given, by using some Euler identities involving harmonic sequences of polynomials.

Pose estimation and object identification using complex algebraic representations

The comparison and alignment of two similar objects is a fundamental problem in pattern recognition and computer vision that has been considered using various approaches. In this work, we employ a complex representation for an algebraic curve, and illustrate how the algebraic transformation which relates two Euclidean equivalent curves can be determined using this representation. The idea is based on a complex representation of 2D points expressed in terms of the orthogonalx andy variables, with rotations of the complex numbers described using Euler's identity. We develop a simple formula for integer multiples of the rotation angle of the Euclidean transformation in terms of the real coefficients of implicit polynomial equations that are used to model 2D free-form objects. When there is a translation, it can be determined using some new results on the conic-line factors of implicit polynomial curves. Experimental results are presented for data sets characterised by both noisy and missing data points to illustrate and validate our procedures.

A generalization of Sylvester's identity

We consider a new generalization of Euler's and Sylvester's identities for partitions. Our proof is based on an explicit bijection.

A Higher-Level Bailey Lemma

We propose a generalization of Bailey's lemma, useful for proving q-series identities. As an application, generalizations of Euler's identity, the Rogers–Ramanujan identities, and the Andrews–Gordon identities are derived. This generalized Bailey lemma also allows one to derive the branching functions of higher-level [Formula: see text] cosets.

Algebraic Dilogarithm Identities

The Rogers L-function $$L(x) = \sum\limits_{n = 1}^\infty {\frac{{x^n }} {{n^2 }} + \frac{1} {2}\log x} \log (1 - x) $$ satisfies the functional equation $$L(x) + L(y) = L(xy) + L\left( {\frac{{x(1 - y)}} {{1 - xy}}} \right) + L\left( {\frac{{y(1 - x)}} {{1 - xy}}} \right) $$ .From this we derive several other such equations, including Euler's identity L(x)+L(1-x)=L(1) and various identities arising from summation and transformation formulas for basic hypergeometric series. We also obtain some new equations of the form $$\sum\limits_{k = 0}^n {c_k L(\theta ^k ) = 0} $$ where θ is algebraic and the c k are integers.