Chapter 7: Applications of the residue theorem

Question 1

Cauchy PV of th eintegral

$$\int_\infty^\infty f(x) dx $$

Answer 1

P.V. $\int_\infty^\infty f(x) dx = \lim_{R\to\infty} \int_R^R f(x) dx$
provided that this limit exists

Question 2

rel betwen $$\int_\infty^\infty f(x) dx $$ and pv cauchy

Answer 2

If the intergral converges on $-\infty \to +\infty$ (i.e. on 0 to infty and on neg then) then it will equal cauchy pv

converse not true in general

Question 3

Lemma

when does func (which 2 conditions) $\int_\infty^\infty f(x) dx $ conv and = cpv of integral

Answer 3

f is cont

even function on $(-\infty , +\infty)$