| log(1 + x) = x - | x² | + | x³ | - . . . x² < 1 |
| 2 | 3 |
It is a solution of the differential equation Dₓy = 1/y. If y is the logarithm of x, then x is the antilogarithm of y. The logarithm to the base 10 of x, log₁₀ x, equals the logarithm to the base e of x, logₑ x, divided by logₑ 10. See also textbooks on algebra and calculus.
sine, cosine, tangent, antitangent
These also are important mathematical functions. The sine and cosine are solutions of the differential equation Dₓ(Dₓy) =-y and are written as sin x and cos x. They can be computed from
| sin x = x - | x³ | + | x⁵ | - . . . |
| 1 · 2· 3 | 1 · 2 · 3· 4· 5 | |||
| cos x = 1 - | x² | + | x₄ | - . . . |
| 1 · 2 | 1 · 2 · 3· 4 | |||
The tangent of x is simply sine of x divided by cosine of x. If y is the tangent of x, then x is the antitangent of y. See also references on trigonometry and on calculus. Trigonometric tables include sine, cosine, tangent, and related functions.
Bessel functions
These are mathematical functions that were named after Friedrich W. Bessel, a Prussian astronomer who lived from 1784 to 1846. Bessel functions are found as some of the solutions of the differential equation
x² Dₓ(Dₓy) + x Dₓy + (x² - n²)y = O
This equation arises in a number of physical problems in the fields of electricity, sound, heat flow, air flow, etc.