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Particular values of the gamma function. The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
The gamma function is the unique function that simultaneously satisfies. , for all complex numbers except the non-positive integers, and, for integer n, for all complex numbers . [1] In a certain sense, the log-gamma function is the more natural form; it makes some intrinsic attributes of the function clearer.
A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple ...
30° and 60°. The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained.
Starting at 0, add 1 for each cell whose distance to the origin (0,0) is less than or equal to r. When finished, divide the sum, representing the area of a circle of radius r, by r2 to find the approximation of π. For example, if r is 5, then the cells considered are: (−5,5) (−4,5)
Babylonian mathematics (also known as Assyro-Babylonian mathematics) [1][2][3][4] is the mathematics developed or practiced by the people of Mesopotamia, as attested by sources mainly surviving from the Old Babylonian period (1830–1531 BC) to the Seleucid from the last three or four centuries BC. With respect to content, there is scarcely any ...
John Wallis built upon this work by considering expressions of the form y = (1 − x 2) m where m is a fraction. He found that (written in modern terms) the successive coefficients c k of (− x 2 ) k are to be found by multiplying the preceding coefficient by m − ( k − 1) / k (as in the case of integer exponents), thereby ...
The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial