Analysis III TT23, Cauchy principal value
Flashcards
Explain why the improper integral
\[\int^1 _ {-1} \frac 1 x \text{ d}x \]
is undefined, but
\[\text{PV }\int^1 _ {-1} \frac 1 x \text{ d}x \]
is.
This is equivalent to the limit
\[\lim _ {\varepsilon' \to 0, \varepsilon \to 0} \log \frac{\varepsilon'}{\varepsilon}\]If both $\varepsilon$ are the same, then the limit exists, otherwise it doesn’t.