His dissertation and much of his later work dealt with the solutions of ordinary differential equations and the associated expansions of arbitrary functions.
(10)D'Alembert showed that the general solution to (10) isy x, t) = f(x + ct) + g(x − ct) (11 where f and g are arbitrary functions (of one variable).
For given let be arbitrary functions in.
where, and are arbitrary functions of while, and are arbitrary functions of and.
Legendre wavelets produce a good approximation of arbitrary functions.
where, and are arbitrary functions of their arguments.
To prove this property, we fix arbitrary functions with.
holds for arbitrary functions {g k }k∈ℤon ℝn+1.
All material properties are arbitrary functions along the beam thickness.
It applies to classes of differential systems whose arbitrary functions involve all equations' independent variables.
Relatedly, an important approach to objective probability that is gaining in popularity involves the so-called method of arbitrary functions.
