mathematical identities pdf

Revised Edition: 2016 ISBN 978-1-283-49630-8 © All rights reserved Published by: Research World 48 West 48 Street, Suite 1116, New York, NY 10036, United States Email: info@wtbooks.com  www.TechnicalBooksPDF.com Table of Contents Introduction Chapter - Capelli's Identity Chapter - Binet–Cauchy Identity, Brahmagupta–Fibonacci Identity and Green's Identities Chapter - Difference of Two Squares, Euler's Identity and Jacobi Triple Product Chapter - Differentiation Rules Chapter - Abel's Identity and Morrie's Law Chapter - Lagrange's Identity (Boundary Value Problem) and Liouville's Formula Chapter - Newton's Identities Chapter - Lagrange's Identity and Polarization Identity Chapter - Pascal's Rule, Polynomial Identity Ring and q-Vandermonde Identity Chapter 10 - Pythagorean Trigonometric Identity Chapter 11 - Squared Triangular Number, Tangent half-angle Formula and Vandermonde's Identity Chapter 12 - Vector Calculus Identities WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Introduction In mathematics, the term identity has several different important meanings: • An identity is a relation which is tautologically true This is usually taken to mean something that is true by definition, either directly by the definition, or as a consequence of it For example, algebraically, this occurs if an equation is satisfied for all values of the involved variables Definitions are often indicated by the 'triple bar' symbol ≡, such as A2 ≡ x·x The symbol ≡ can also be used with other meanings, but these can usually be interpreted in some way as a definition, or something which is otherwise tautologically true (for example, a congruence relation) • In algebra, an identity or identity element of a set S with a binary operation · is an element e that, when combined with any element x of S, produces that same x That is, e·x = x·e = x for all x in S An example of this is the identity matrix • The identity function from a set S to itself, often denoted id or idS, is the function which maps every element to itself In other words, id(x) = x for all x in S This function serves as the identity element in the set of all functions from S to itself with respect to function composition Examples Identity relation A common example of the first meaning is the trigonometric identity which is true for all complex values of θ (since the complex numbers sin and cos), as opposed to are the domain of which is true only for some values of θ, not all For example, the latter equation is true when false when WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Identity element The concepts of "additive identity" and "multiplicative identity" are central to the Peano axioms The number is the "additive identity" for integers, real numbers, and complex numbers For the real numbers, for all and Similarly, The number is the "multiplicative identity" for integers, real numbers, and complex numbers For the real numbers, for all and Identity function A common example of an identity function is the identity permutation, which sends each element of the set to itself or to itself in natural order Comparison These meanings are not mutually exclusive; for instance, the identity permutation is the identity element in the group of permutations of under composition WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Chapter-1 Capelli's Identity In mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process Statement Suppose that xij for i,j = 1, ,n are commuting variables Write Eij for the polarization operator The Capelli identity states that the following differential operators, expressed as determinants, are equal: Both sides are differential operators The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column It can be formally written as WORLD TECHNOLOGIES www.TechnicalBooksPDF.com where in the product first come the elements from the first column, then from the second and so on The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant The operators Eij can be written in a matrix form: E = XDt, where E,X,D are matrices with elements Eij, xij, respectively If all elements in these matrices would be commutative then clearly det(E) = det(X)det(Dt) The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above The only price for the noncommutivity is a small correction: (n − i)δij on the left hand side For generic noncommutative matrices formulas like det(AB) = det(A)det(B) not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices That is why the Capelli identity still holds some mystery, despite many proofs offered for it A very short proof does not seem to exist Direct verification of the statement can be given as an exercise for n' = 2, but is alraeady long for n = Relations with representation theory Consider the following slightly more general context Suppose that n and m are two integers and xij for i = 1, ,n,j = 1, ,m, be commuting variables Redefine Eij by almost the same formula: with the only difference that summation index a ranges from to m One can easily see that such operators satisfy the commutation relations: Here [a,b] denotes the commutator ab − ba These are the same commutation relations which are satisfied by the matrices eij which have zeros everywhere except the position (i,j), where stands (eij are sometimes called matrix units) Hence we conclude that the correspondence defines a representation of the Lie algebra in the vector space of polynomials of xij WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Case m = and representation Sk Cn It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi: In particular, for the polynomials of the first degree it is seen that: Hence the action of Eij restricted to the space of first-order polynomials is exactly the same as the action of matrix units eij on vectors in So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentation of the Lie algebra , which we identified with the standard representation in Going further, it is seen that the differential operators Eij preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentation of the Lie algebra One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power of the standard representation One can also easily identify the highest weight structure of these representations The monomial is a highest weight vector, indeed: equals to (k, 0, ,0), indeed: for i < j Its highest weight Such representation is sometimes called bosonic representation of Similar formulas define the so-called fermionic representation, here ψi are anti-commuting variables Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to i.e anti-symmetric tensor power of Highest weight of such representation is (0, , 0, 1, 0, , 0) These representations for k = 1, , n are fundamental representations of Capelli identity for m = Let us return to the Capelli identity One can prove the following: the motivation for this equality is the following: consider for some commuting variables xi,pj The matrix Ec is of rank one and hence its determinant is equal to WORLD TECHNOLOGIES www.TechnicalBooksPDF.com zero Elements of matrix E are defined by the similar formulas, however, its elements not commute The Capelli identity shows that the commutative identity: det(Ec) = can be preserved for the small price of correcting matrix E by (n − i)δij Let us also mention that similar identity can be given for the characteristic polynomial: where The commutative counterpart of this is a simple fact that for rank = matrices the characteristic polynomial contains only the first and the second coefficients Let us consider an example for n = Using we see that this is equal to: The universal enveloping algebra and its center An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is the commutator [Eij,det(E + (n − i)δij)] = is equal to zero It can be generalized: Consider any elements Eij in any ring, such that they satisfy the commutation relation [Eij,Ekl] = δjkEil − δilEkj, (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com where then: • elements Ck commute with all elements Eij • elements Ck can be given by the formulas similar to the commutative case: i.e they are sums of principal minors of the matrix E, modulo the Capelli correction + (k − i)δij In particular element C0 is the Capelli determinant considered above These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the direct few lines short proof does not seem to exist, despite the simplicity of the formulation The universal enveloping algebra can defined as an algebra generated by Eij subject to the relations [Eij,Ekl] = δjkEil − δilEkj alone The proposition above shows that elements Ckbelong to the center of It can be shown that they actually are free generators of the center of They are sometimes called Capelli generators The Capelli identities for them will be discussed below Consider an example for n = It is immediate to check that element (E11 + E22) commute with Eij (It corresponds to an obvious fact that the identity matrix commute with all other matrices) More instructive is to check commutativity of the second element with Eij Let us it for E12: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Chapter-11 Squared Triangular Number, Tangent halfangle Formula and Vandermonde's Identity Squared triangular number Visual demonstration that the square of a triangular number equals a sum of cubes In number theory, the sum of the first n cubes is the square of the nth triangular number That is, This identity is sometimes called Nicomachus's theorem WORLD TECHNOLOGIES www.TechnicalBooksPDF.com History Stroeker (1995), writing about Nicomachus's theorem, claims that "every student of number theory surely must have marveled at this miraculous fact" While Stroeker's statement may perhaps be a poetic exaggeration, it is true that many mathematicians have studied this equality and have proven it in many different ways Pengelley (2002) finds references to the identity in several ancient mathematical texts: the works of Nicomachus in what is now Jordan in the first century CE, Aryabhata in India in the fifth century, and Al-Karaji circa 1000 in Persia Bressoud (2004) mentions several additional early mathematical works on this formula, by Alchabitius (tenth century Arabia), Gersonides (circa 1300 France), and Nilakantha Somayaji (circa 1500 India); he reproduces Nilakantha's visual proof Numeric values; geometric and probabilistic interpretation The sequence of squared triangular numbers is 0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, (sequence A000537 in OEIS) These numbers can be viewed as figurate numbers, a four-dimensional hyperpyramidal generalization of the triangular numbers and square pyramidal numbers As Stein (1971) observes, these numbers also count the number of rectangles with horizontal and vertical sides formed in an n×n grid For instance, the points of a 4×4 grid can form 36 different rectangles The number of squares in a square grid is similarly counted by the square pyramidal numbers The identity also admits a natural probabilistic interpretation as follows Let X,Y,Z,W be four integer numbers independently and uniformly chosen at random between and n Then, the probability that W be not less than any other is equal to the probability that both Y be not less than X and W be not less than Z, that is, Indeed, these probabilities are respectively the left and right sides of the Nichomacus identity, normalized over n4 Proofs Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers: + + 27 + 64 + 125 + = (1) + (3 + 5) + (7 + + 11) + (13 + 15 + 17 + 19) + (21 + 23 + 25 + 27 + 29) + = + + + + + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 WORLD TECHNOLOGIES www.TechnicalBooksPDF.com The sum of any set of consecutive odd numbers starting from is a square, and the quantity that is squared is the count of odd numbers in the sum The latter is easily seen to be a count of the form 1+2+3+4+ +n In the more recent mathematical literature, Stein (1971) uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity; he observes that it may also be proved easily (but uninformatively) by induction, and states that Toeplitz (1963) provides "an interesting old Arabic proof" Kanim (2004) provides a purely visual proof, Benjamin and Orrison (2002) provide two additional proofs, and Nelsen (1993) gives seven geometric proofs Generalizations A similar result to Nicomachus's theorem holds for all power sums, namely that odd power sums (sums of odd powers) are a polynomial in triangular numbers These are called Faulhaber polynomials, of which the sum of cubes is the simplest and most elegant example Stroeker (1995) studies more general conditions under which the sum of a consecutive sequence of cubes forms a square Garrett and Hummel (2004) and Warnaar (2004) study polynomial analogues of the square triangular number formula, in which series of polynomials add to the square of another polynomial Tangent half-angle formula In trigonometry, the tangent half-angle formulas relate the tangent of one half on an angle to trigonometric functions of the entire angle, as follows: Variations on this theme include the following identities: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com A geometric proof Geometry of the tangent half-angle formula The Weierstrass substitution In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t These identities are known collectively as the tangent half-angle formulae because of the definition of t These identities can be useful in calculus for converting rational functions in sine and cosine to functions of t in order to find their antiderivatives Technically, the existence of the tangent half-angle formulae stems from the fact that the circle is an algebraic curve of genus One then expects that the 'circular functions' should be reducible to rational functions Geometrically, the construction goes like this: for any point (cos φ, sin φ) on the unit circle, draw the line passing through it and the point (−1,0) This point crosses the y-axis at some point y = t One can show using simple geometry that t = tan(φ/2) The equation for the drawn line is y = (1 + x)t The equation for the intersection of the line and circle is then a quadratic equation involving t The two solutions to this equation are (−1, 0) and (cos φ, sin φ) This allows us to write the latter as rational functions of t (solutions are given below) Note also that the parameter t represents the stereographic projection of the point (cos φ, sin φ) onto the y-axis with the center of projection at (−1,0) Thus, the tangent half-angle formulae give conversions between the stereographic coordinate t on the unit circle and the standard angular coordinate φ WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Then we have and By eliminating phi between the directly above and the initial definition of t, one arrives at the following useful relationship for the arctangent in terms of the natural logarithm In calculus, the Weierstrass substitution is used to find antiderivatives of rational functions of sin(φ) and cos(φ) After setting This implies that and therefore Hyperbolic identities One can play an entirely analogous game with the hyperbolic functions A point on (the right branch of) a hyperbola is given by (cosh θ, sinh θ) Projecting this onto y-axis from the center (−1, 0) gives the following: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com with the identities and The use of this substitution for finding antiderivatives was introduced by Karl Weierstrass Finding θ in terms of t leads to following relationship between the hyperbolic arctangent and the natural logarithm: The Gudermannian function Comparing the hyperbolic identities to the circular ones, one notices that they involve the same functions of t, just permuted If we identify the parameter t in both cases we arrive at a relationship between the circular functions and the hyperbolic ones That is, if then The function gd(θ) is called the Gudermannian function The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that WORLD TECHNOLOGIES www.TechnicalBooksPDF.com does not involve complex numbers The above descriptions of the tangent half-angle formulae (projection the unit circle and standard hyperbola onto the y-axis) give a geometric interpretation of this function Vandermonde's identity In combinatorics, Vandermonde's identity, or Vandermonde's convolution, named after Alexandre-Théophile Vandermonde (1772), states that for binomial coefficients This identity was given already in 1303 by the Chinese mathematician Zhu Shijie (Chu Shi-Chieh) There is a q-analog to this theorem called the q-Vandermonde identity Algebraic proof In general, the product of two polynomials with degrees m and n, respectively, is given by where we use the convention that = for all integers i > m and bj = for all integers j > n By the binomial theorem, Using the binomial theorem also for the exponents m and n, and then the above formula for the product of polynomials, we obtain WORLD TECHNOLOGIES www.TechnicalBooksPDF.com where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively By comparing coefficients of xr, Vandermonde's identity follows for all integers r with ≤ r ≤ m + n For larger integers r, both sides of Vandermonde's identity are zero due to the definition of binomial coefficients Combinatorial proof Vandermonde's identity also admits a more combinatorics-flavored double counting proof, as follows Suppose a committee in the US Senate consists of m Democrats and n Republicans In how many ways can a subcommittee of r members be formed? The answer is of course But on the other hand, the answer is the sum over all possible values of k, of the number of subcommittees consisting of k Democrats and r − k Republicans Generalized Vandermonde's identity If in the algebraic derivation above more than two polynomials are used, it results in the generalized Vandermonde's identity For y + polynomials: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com The hypergeometric probability distribution When both sides have been divided by the expression on the left, so that the sum is 1, then the terms of the sum may be interpreted as probabilities The resulting probability distribution is the hypergeometric distribution That is the probability distribution of the number of red marbles in r draws without replacement from an urn containing n red and m blue marbles Chu–Vandermonde identity The identity generalizes to non-integer arguments In this case, it is known as the Chu– Vandermonde identity and takes the form for general complex-valued s and t and any non-negative integer n This identity may be re-written in terms of the falling Pochhammer symbols as in which form it is clearly recognizable as an umbral variant of the binomial theorem The Chu–Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that where is the hypergeometric function and Γ(n + 1) = n! is the gamma function One regains the Chu–Vandermonde identity by taking a = −n and applying the identity liberally WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Chapter-12 Vector Calculus Identities The following identities are important in vector calculus: Single operators (summary) This section explicitly lists what some symbols mean for clarity Divergence Divergence of a vector field For a vector field , divergence is generally written as and is a scalar Divergence of a tensor For a second order tensor , divergence is generally written as and is a vector More generally speaking, the divergence of a tensor of order n is a contraction to a tensor of order n-1 Curl For a vector field , curl is generally written as: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com and is a vector field Gradient Gradient of a vector field For a vector field , gradient is generally written as: and is a tensor Gradient of a scalar field For a scalar field, ψ, the gradient is generally written as and is a vector Combinations of multiple operators Curl of the gradient The curl of the gradient of any scalar field is always the zero vector: One way to establish this identity is to use three-dimensional Cartesian coordinates According to curl, where the right hand side is a determinant, and i, j, k are unit vectors pointing in the positive axes directions, and ∂x = ∂ / ∂ x etc For example, the x-component of the above equation is: WORLD TECHNOLOGIES www.TechnicalBooksPDF.com where the left-hand side equals zero due to the equality of mixed partial derivatives Divergence of the curl The divergence of the curl of any vector field A is always zero: Divergence of the gradient The Laplacian of a scalar field is defined as the divergence of the gradient: Note that the result is a scalar quantity Curl of the curl Here, ∇2 is the vector Laplacian operating on the vector field A Properties Distributive property Vector dot product In simpler form, using Feynman subscript notation: where the notation ∇A means the subscripted gradient operates on only the factor A A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed The above identity is then expressed as: where overdots define the scope of the vector derivative In the first term it is only the first (dotted) factor that is differentiated, while the second is held constant Likewise, in WORLD TECHNOLOGIES www.TechnicalBooksPDF.com the second term it is the second (dotted) factor that is differentiated, and the first is held constant As a special case, when A = B: Vector cross product where the Feynman subscript notation ∇B means the subscripted gradient operates on only the factor B In overdot notation, explained above: Product of a scalar and a vector Product rule for the gradient The gradient of the product of two scalar fields ψ and φ follows the same form as the product rule in single variable calculus Summary of all identities Addition and multiplication • • • • • • • ( scalar triple product) (vector triple product) WORLD TECHNOLOGIES www.TechnicalBooksPDF.com • • Differentiation DCG chart: A simple chart depicting all rules pertaining to second derivatives D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively Arrows indicate existence of second derivatives Blue circle in the middle represents curl of curl, whereas the other two red circles(dashed) mean that DD and GG not exist • • • • • • • • • • • • • (scalar Laplacian) (vector Laplacian) • WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Integration (Divergence theorem) • • • • • • (Green's first identity) (Gr een's second identity) (Stokes' theorem) • WORLD TECHNOLOGIES www.TechnicalBooksPDF.com ... chakravala (cyclic) method, was also based on this identity Green's identities In mathematics, Green's identities are a set of three identities in vector calculus They are named after the mathematician... Formula and Vandermonde's Identity Chapter 12 - Vector Calculus Identities WORLD TECHNOLOGIES www.TechnicalBooksPDF.com Introduction In mathematics, the term identity has several... model Permanents, immanants, traces – "higher Capelli identities" The original Capelli identity is a statement about determinants Later, analogous identities were found for permanents, immanants and