order,
n. 1. the number of times a given function must be differentiated to obtain a given derivative. For instance, the third derivative, f''', of a function f is of order 3. 2. the order of the highest order derivative in a differential equation. For example,
is an equation of second order. Compare degree. 3. the position of elements in a sequence. For example, in an ordered set, the order of elements makes a difference, and <a, b> = <b, a> if and only if a = b.4. the number of rows or columns in a square matrix or determinant. 5. the number of elements of a finite group or set; its cardinality. 6. also called period. (for an element a in a group) the smallest number of times the given element has to be multiplied by itself to yield the identity element of the group; the least positive integer n such that an = e, written |a| = n. If <a> is the set generated by a, then |a|=|<a> |. If no such integer exists the element has infinite order.7. the multiplicity of a zero or a pole. 8. the number of poles, counting multiplicity, in any fundamental parallelogram of a doubly periodic function, such as an elliptic function. 9. (for an entire function) the quantity
where m(r) is the maximum modulus of the given entire function on the disk of radius r.
The order and genus, γ, of an entire function satisfy γ ≤ λ ≤ γ + 1. See three-circle theorem
10. an alternative term for ordering.
11. another term for order of magnitude, order of symmetry, or order of convergence.
12. of the order of. a.. having approximately the magnitude of.
b. (of a function) approximating to a constant multiple of another function for large values of the argument. Similarly, one speaks of functions as being of lower order or of higher order than a given function.
c. (of a function) having another function as an asymptote as their arguments tend to infinity, having 1 as the limit of their quotient. See order notation.
is an equation of second order. Compare degree. 3. the position of elements in a sequence. For example, in an ordered set, the order of elements makes a difference, and <a, b> = <b, a> if and only if a = b.4. the number of rows or columns in a square matrix or determinant. 5. the number of elements of a finite group or set; its cardinality. 6. also called period. (for an element a in a group) the smallest number of times the given element has to be multiplied by itself to yield the identity element of the group; the least positive integer n such that an = e, written |a| = n. If <a> is the set generated by a, then |a|=|<a> |. If no such integer exists the element has infinite order.7. the multiplicity of a zero or a pole. 8. the number of poles, counting multiplicity, in any fundamental parallelogram of a doubly periodic function, such as an elliptic function. 9. (for an entire function) the quantity 10. an alternative term for ordering.
11. another term for order of magnitude, order of symmetry, or order of convergence.
12. of the order of. a.. having approximately the magnitude of.
b. (of a function) approximating to a constant multiple of another function for large values of the argument. Similarly, one speaks of functions as being of lower order or of higher order than a given function.
c. (of a function) having another function as an asymptote as their arguments tend to infinity, having 1 as the limit of their quotient. See order notation.