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Auxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.
Auxiliary functions are not a rigorously defined kind of function, rather they are functions which are either explicitly constructed or at least shown to exist and which provide a contradiction to some assumed hypothesis, or otherwise prove the result in question. Creating a function during the course of a proof in order to prove the result is not a technique exclusive to transcendence theory, but the term "auxiliary function" usually refers to the functions created in this area.
Because of the naming convention mentioned above, auxiliary functions can be dated back to their source simply by looking at the earliest results in transcendence theory. One of these first results was Liouville's proof that transcendental numbers exist when he showed that the so called Liouville numbers were transcendental. He did this by discovering a transcendence criterion which these numbers satisfied. To derive this criterion he started with a general algebraic number α and found some property that this number would necessarily satisfy. The auxiliary function he used in the course of proving this criterion was simply the minimal polynomial of α, which is the irreducible polynomial f with integer coefficients such that f(α) = 0. This function can be used to estimate how well the algebraic number α can be estimated by rational numbers p/q. Specifically if α has degree d at least two then he showed that
and also, using the mean value theorem, that there is some constant depending on α, say c(α), such that
Combining these results gives a property that the algebraic number must satisfy; therefore any number not satisfying this criterion must be transcendental.
The auxiliary function in Liouville's work is very simple, merely a polynomial that vanishes at a given algebraic number. This kind of property is usually the one that auxiliary functions satisfy. They either vanish or become very small at particular points, which is usually combined with the assumption that they do not vanish or can't be too small to derive a result.
Another simple, early occurrence is in Fourier's proof of the irrationality of e, though the notation used usually disguises this fact. Fourier's proof used the power series of the exponential function:
By truncating this power series after, say, N + 1 terms we get a polynomial with rational coefficients of degree N which is in some sense "close" to the function ex. Specifically if we look at the auxiliary function defined by the remainder:
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Auxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.
Auxiliary functions are not a rigorously defined kind of function, rather they are functions which are either explicitly constructed or at least shown to exist and which provide a contradiction to some assumed hypothesis, or otherwise prove the result in question. Creating a function during the course of a proof in order to prove the result is not a technique exclusive to transcendence theory, but the term "auxiliary function" usually refers to the functions created in this area.
Because of the naming convention mentioned above, auxiliary functions can be dated back to their source simply by looking at the earliest results in transcendence theory. One of these first results was Liouville's proof that transcendental numbers exist when he showed that the so called Liouville numbers were transcendental. He did this by discovering a transcendence criterion which these numbers satisfied. To derive this criterion he started with a general algebraic number α and found some property that this number would necessarily satisfy. The auxiliary function he used in the course of proving this criterion was simply the minimal polynomial of α, which is the irreducible polynomial f with integer coefficients such that f(α) = 0. This function can be used to estimate how well the algebraic number α can be estimated by rational numbers p/q. Specifically if α has degree d at least two then he showed that
and also, using the mean value theorem, that there is some constant depending on α, say c(α), such that
Combining these results gives a property that the algebraic number must satisfy; therefore any number not satisfying this criterion must be transcendental.
The auxiliary function in Liouville's work is very simple, merely a polynomial that vanishes at a given algebraic number. This kind of property is usually the one that auxiliary functions satisfy. They either vanish or become very small at particular points, which is usually combined with the assumption that they do not vanish or can't be too small to derive a result.
Another simple, early occurrence is in Fourier's proof of the irrationality of e, though the notation used usually disguises this fact. Fourier's proof used the power series of the exponential function:
By truncating this power series after, say, N + 1 terms we get a polynomial with rational coefficients of degree N which is in some sense "close" to the function ex. Specifically if we look at the auxiliary function defined by the remainder: