Principle of permanence

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The function in the middle is given by x2 sin(1/x) for x not equal to 0 and given by 0 for x=0. This cannot be an analytic function, because it is has infinitely many zeros, but is not itself the zero function.

In mathematics, the principle of permanence is that a complex function (or functional equation) which is 0 on a set with a non-isolated point is 0 everywhere (or at least on the connected component of its domain which contains the point). There are various statements of the principle, depending on the type of function or equation considered.

For a complex function of one variable[edit]

For one variable, the principle of permanence states that if f(z) is an analytic function defined on an open connected subset U of the complex numbers C, and there exists a convergent sequence {an} having a limit L which is in U, such that f(an) = 0 for all n, then f(z) is uniformly zero on U.[1]

For a complex function of two variables[edit]

For a function of two variables, the principle of permanence says that if f(z,w) is an analytic function defined on an open connected subset U of the complex numbers, there exists a convergent sequence {an} having a limit L which is in U, such that f(an) = 0 for all n, then f(z) is uniformly zero on U.[2]

For functional equations involving complex functions[edit]

For a functional equation of the form F(z,f1,...,fn)=0, where the fi are complex functions, the principle of permanence says that any solution to the functional equation remains a solution when we analytically continue each fi along the same curves.[3]

Applications[edit]

One of the main uses of the principle of permanence is to show that a functional equation that holds for the real numbers also holds for the complex numbers.[4]

As an example, the function es+t-eset=0 on the real numbers. By the principle of permanence for functions of two variables, this implies that es+t-eset for all complex numbers, thus proving one of the laws of exponents for complex exponents.[2]

See also[edit]

References[edit]

  1. ^ 'The Language of Science, Tobias Dantzig, Joseph Mazur, and Barry Mazur, 2007, Penguin Books, pp. 98, 212.
  2. ^ a b Gamelin, T. Complex Analysis, UTM Series, Springer-Verlag, 2001c
  3. ^ Gonzales, M. Complex Analysis: Selected Topics, CRC Press, Chapman & Hall/CRC Pure and Applied Mathematics (Book 152), 1991
  4. ^ Dauben, Joseph W. (1979), Georg Cantor: his mathematics and philosophy of the infinite, Boston: Harvard University Press, ISBN 978-0-691-02447-9 .

External links[edit]