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The actual approach appears to have been developed by Clebsch in 1862. [2] Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression, [ 3 ] to Timoshenko beams , [ 4 ] to elastic foundations , [ 5 ] and to problems in which the bending and shear stiffness changes discontinuously in a beam.
Specific choices of give different types of Riemann sums: . If = for all i, the method is the left rule [2] [3] and gives a left Riemann sum.; If = for all i, the method is the right rule [2] [3] and gives a right Riemann sum.
In mathematics, the Riemann–Liouville integral associates with a real function: another function I α f of the same kind for each value of the parameter α > 0.The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of α, I α f is an iterated antiderivative of f of order α.
A vector v (red) represented by • a vector basis (yellow, left: e 1, e 2, e 3), tangent vectors to coordinate curves (black) and • a covector basis or cobasis (blue, right: e 1, e 2, e 3), normal vectors to coordinate surfaces (grey) in general (not necessarily orthogonal) curvilinear coordinates (q 1, q 2, q 3). The basis and cobasis do ...
In 1973, Montgomery & Vaughan reported several generalizations of Hilbert's inequality, considering the bilinear forms ¯ and ¯, where x 1,x 2,...,x m are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the quotient group R/Z) and λ 1,...,λ m are distinct real numbers.
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form () (,), where < (), < and the integrands are functions dependent on , the derivative of this integral is expressible as (() (,)) = (, ()) (, ()) + () (,) where the partial derivative indicates that inside the integral, only the ...
The asymptotic formula for Ai(x) is still valid in the complex plane if the principal value of x 2/3 is taken and x is bounded away from the negative real axis. The formula for Bi( x ) is valid provided x is in the sector x ∈ C : | arg ( x ) | < π 3 − δ {\displaystyle x\in \mathbb {C} :\left|\arg(x)\right|<{\tfrac {\pi }{3}}-\delta ...
Hadamard's gamma function plotted over part of the real axis. Unlike the classical gamma function, it is holomorphic; there are no poles. In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function).