Concave function Concave function In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex. A real-valued function formula_1 on an interval (or, more generally, a convex set in vector space) is said to be "concave" if, for any formula_2 and formula_3 in the interval and for any formula_4, A function is called "strictly concave" if for any formula_7 and formula_8. For a function formula_9, this second definition merely states that for every formula_10 strictly between formula_2 and formula_3, the point formula_13 on
Luneburg lens one side, and formula_14 only on the other: The parameter formula_19 is a constant for any given ray, but differs between rays passing at different distances from the centre of the lens. For rays passing through the centre, it is zero. In some special cases, such as for Maxwell's fish-eye, this first order equation can be further integrated to give a formula for formula_14 as a function or formula_13. In general it provides the relative rates of change of formula_14 and formula_13, which may be integrated numerically to follow the path of the ray through the lens. Luneburg lens A