Cauchy's main value

As Cauchy principal value (after Augustin-Louis Cauchy in) refers mathematical sub-region of the Analysis of the value which is a divergent integral can be assigned when divergent parts of different sign cancel each other.

definition

Cauchy's principal value is a value that can be assigned to 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 . If the limit value then exists ${\displaystyle -\infty \leq a<b\leq \infty }$ ${\displaystyle c\in {]a,b[}}$ ${\displaystyle f\colon {]a,c[}\cup {]c,b[}\to \mathbb {R} }$

${\displaystyle \operatorname {CH} \int _{a}^{b}f(x)\,\mathrm {d} x:=\lim _{\epsilon \rightarrow 0^{+}}\left(\int _{a}^{c-\epsilon }f(x)\,\mathrm {d} x+\int _{c+\epsilon }^{b}f(x)\,\mathrm {d} x\right)\,,}$

this is what Cauchy's principal value is called. ${\displaystyle \operatorname {CH} \int _{a}^{b}f(x)\,\mathrm {d} x}$

• Let be a continuous function , then the limit, if it exists, is called ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle \operatorname {CH} \int _{-\infty }^{\infty }f(x)\,\mathrm {d} x:=\lim _{R\rightarrow \infty }\int _{-R}^{R}f(x)\,\mathrm {d} x}$

also Cauchy's main value.

It is also common to write “VP” (from the French valeur principale ) or “PV” (from the English principal value ) instead of “CH”.

Relationship between Cauchy's principal value and improper integral

If there is an integral over in the improper sense , then Cauchy's main value always exists (according to the second definition) and these two values ​​agree. The existence of the improper integral does not yet follow from the existence of Cauchy's principal value. ${\displaystyle \mathbb {R} }$

Example (CH 1 / x)

Cauchy's principal value - example

The definite integral is examined. The integrand is not defined for (an inner point of the integration area ). This integral is therefore improper in . The antiderivative of the integrand is (see table of derivative and antiderivative functions ). ${\displaystyle \textstyle \int _{-1}^{1}{\frac {1}{x}}\,\mathrm {d} x}$ ${\displaystyle x=0}$ ${\displaystyle ]-1,1[}$ ${\displaystyle 0}$ ${\displaystyle {\tfrac {1}{x}}}$ ${\displaystyle \ln \left|x\right|}$

${\displaystyle {\begin{aligned}\Rightarrow \int _{-1}^{1}{\frac {1}{x}}\,\mathrm {d} x=&\int _{-1}^{0}{\frac {1}{x}}\,\mathrm {d} x+\int _{0}^{1}{\frac {1}{x}}\mathrm {d} x={\Big [}\ln \left|x\right|{\Big ]}_{x=-1}^{0}+{\Big [}\ln \left|x\right|{\Big ]}_{x=0}^{1}\\=&\lim _{x\rightarrow 0^{-}}\ln \left|x\right|-\ln \left|-1\right|+\ln \left|1\right|-\lim _{x\rightarrow 0^{+}}\ln \left|x\right|=-\infty -\left(-\infty \right)\end{aligned}}}$

This integral does not exist as an improper Riemann integral, but the main Cauchy value is : ${\displaystyle 0}$

${\displaystyle \operatorname {CH} \int _{-1}^{1}{\frac {1}{x}}\,\mathrm {d} x=\lim _{\epsilon \rightarrow 0^{+}}\left(\int _{-1}^{0-\epsilon }{\frac {1}{x}}\,\mathrm {d} x+\int _{0+\epsilon }^{1}{\frac {1}{x}}\,\mathrm {d} x\right)=\lim _{\epsilon \rightarrow 0^{+}}(\ln(\epsilon )-\ln(\epsilon ))=0}$

The Cauchy main value makes it possible to assign a value to an integral that does not exist in either the Riemannian or Lebesgueian sense .

If is continuous on the real axis and only differs from zero on a limited interval, then the expression in particular exists . This means that like the delta distribution , it can also be understood as a distribution . ${\displaystyle f}$ ${\displaystyle \textstyle \operatorname {CH} \int _{-\infty }^{+\infty }f(x){\frac {1}{x}}\,\mathrm {d} x}$ ${\displaystyle \operatorname {CH} {\tfrac {1}{x}}}$

Substitution i. General not allowed

However, the main value of an integral generally does not remain invariant under substitution . If, for example, the function is defined by for and for , then the substitution rule applies ${\displaystyle \varphi }$ ${\displaystyle \varphi (x)=x^{3}}$ ${\displaystyle x\leq 0}$ ${\displaystyle \varphi (x)=x^{2}}$ ${\displaystyle x\geq 0}$

${\displaystyle \int _{\varphi (a)}^{\varphi (b)}{\frac {1}{t}}\,\mathrm {d} t=\int _{a}^{b}{\frac {1}{\varphi (x)}}\varphi '(x)\,\mathrm {d} x}$

whenever or applies. For , however, the main value of the left integral is a finite number, but the main value of the right integral is : ${\displaystyle 0<a\leq b}$ ${\displaystyle a\leq b<0}$ ${\displaystyle a<0<b}$ ${\displaystyle -\infty }$

${\displaystyle \operatorname {CH} \int _{a^{3}}^{b^{2}}{\frac {1}{t}}\,\mathrm {d} t=\ln {\biggl |}{\frac {b^{2}}{a^{3}}}{\biggr |}}$

${\displaystyle \operatorname {CH} \int _{a}^{b}{\frac {\varphi '(x)}{\varphi (x)}}\,\mathrm {d} x=\lim _{\varepsilon \to 0+}{\biggl (}\int _{a}^{-\varepsilon }{\frac {3x^{2}}{x^{3}}}\,\mathrm {d} x+\int _{\varepsilon }^{b}{\frac {2x}{x^{2}}}\,\mathrm {d} x{\biggr )}=\lim _{\varepsilon \to 0+}{\biggl (}\ln {\biggl |}{\frac {b^{2}}{a^{3}}}{\biggr |}+\ln \varepsilon {\biggr )}=-\infty }$

Web links

• Eric W. Weisstein : Cauchy Principal Value . In: MathWorld (English).

Individual evidence

1. ↑ Klaus Fritzsche: Basic Course Function Theory: An introduction to complex analysis and its applications. 1st edition, Spektrum Akademischer Verlag, ISBN 3827419492 , p. 155.
2. ↑ Eberhard Freitag , Rolf Busam: Function theory. Springer-Verlag, Berlin, ISBN 3-540-67641-4 , p. 177.
3. ↑ Cauchy's main value . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
4. ↑ Eberhard Freitag , Rolf Busam: Function theory. Springer-Verlag, Berlin, ISBN 3-540-67641-4 , pp. 177-178.