Cauchy principal value

As a Cauchy principal value (after Augustin- Louis Cauchy ) is called the value that you can assign to a divergent integral when divergent parts of opposite sign cancel each other in the mathematical subfield of Analysis.

Definition

The Cauchy principal value is a value that you can assign certain divergent integrals. There are two different cases in which one speaks of a Cauchy principal value.

  • Let and be a real number. The function is riemann integrable. Then there exists the limit
  • Is a continuous function, the limit is, if it exists,

It is also common, "VP " (from the French valeur principale ), or "PV " (from the Engl principal value) rather than "CH" to write.

Relationship between cauchyschem principal value integral and uneigentlichem

If there is an integral over in the unqualified sense, so there is also always the Cauchy principal value ( according to the second definition ) and these two values ​​match. From the other hand, the existence of the Cauchy principal value does not follow from the existence of the improper integral.

Example ( CH 1 / x)

It is the definite integral studied. The integrand is not defined ( an interior point of the integration range). Thus, this integral is improper in. The antiderivative of the integrand (see table of derivation and primitive functions ).

This integral thus does not exist as an improper Riemann integral, the Cauchy principal value is however:

The Cauchy principal value thus makes it possible to assign a value to an integral that involves neither exists in the Riemannian sense in the Lebesgue sense.

If it is different to the real axis and continuously only on a limited interval of zero, so there is the particular expression. This means that as the delta function can also be understood as a distribution.

Substitution generally involves. not allowed

However, the main value of an integral is not invariant under substitution in general. Although, if one defines about the function by for and for, then by the substitution rule

Whenever or applies. For, however, the principal value of an integral of a finite number of major value of the second integral, however, is:

Mathematical analysis
170069
de