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The above have been generalized to sums of N exponentials [15] with increasing accuracy in terms of N so that erfc x can be accurately approximated or bounded by 2Q̃(√ 2 x), where ~ = =. In particular, there is a systematic methodology to solve the numerical coefficients {( a n , b n )} N
List of effects; List of forms of electricity named after scientists; List of engineering blunders; Eppendorf & Science Prize for Neurobiology; List of scientific equations named after people; List of types of equilibrium; List of members of the European Academy of Sciences and Arts; List of existing technologies predicted in science fiction ...
As (+) = and (+) + =, the sum and the product of conjugate expressions do not involve the square root anymore. This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation).
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
Science (from Latin scientia, meaning "knowledge") is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. [1] [2] The branches of science, also referred to as scientific fields, scientific disciplines, or just sciences, can be arbitrarily divided into three major ...
One can prove [citation needed] that = is the largest possible number for which the stopping criterion | + | < ensures ⌊ + ⌋ = ⌊ ⌋ in the algorithm above.. In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than 1 should be used to protect against round-off errors.
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Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.