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The product-to-sum identities [28] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. [29]
Illustration of the sum formula. Draw a horizontal line (the x -axis); mark an origin O. Draw a line from O at an angle α {\displaystyle \alpha } above the horizontal line and a second line at an angle β {\displaystyle \beta } above that; the angle between the second line and the x -axis is α + β {\displaystyle \alpha +\beta } .
The map from the sum to the homology group of the product is called the cross product. More precisely, there is a cross product operation by which an i -cycle on X and a j -cycle on Y can be combined to create an ( i + j ) {\displaystyle (i+j)} -cycle on X × Y {\displaystyle X\times Y} ; so that there is an explicit linear mapping defined from ...
The proof (Todhunter, [1] Art.49) of the first formula starts from the identity = , using the cosine rule to express A in terms of the sides and replacing the sum of two cosines by a product. (See sum-to-product identities.) The second formula starts from the identity = + , the third is a quotient and the remainder follow by applying the ...
The distribution of the product of correlated non-central normal samples was derived by Cui et al. [11] and takes the form of an infinite series of modified Bessel functions of the first kind. Moments of product of correlated central normal samples. For a central normal distribution N(0,1) the moments are
The left equality comes from applying the distributive law to the product and recognizing the resulting terms as the prime factorizations of the terms in the harmonic series, and the right equality uses the standard formula for a geometric series. The product is divergent, just like the sum, but if it converged one could take logarithms and ...
By the fundamental theorem of arithmetic, the partial product when expanded out gives a sum consisting of those terms n −s where n is a product of primes less than or equal to q. The inequality results from the fact that therefore only integers larger than q can fail to appear in this expanded out partial product.
The formula for an integration by parts is () ′ = [() ()] ′ (). Beside the boundary conditions , we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ...