calculate-dx-x-2-z-with-z-from-C-

Question Number 67850 by mathmax by abdo last updated on 01/Sep/19

c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{d x}{x^{2} - z} w i t h z f r o m C

Commented by mathmax by abdo last updated on 01/Sep/19

l e t I = \int_{- \infty}^{+ \infty} \frac{d x}{x^{2} - z} l e t z = {r e}^{i θ} \Rightarrow I = \int_{- \infty}^{+ \infty} \frac{d x}{x^{2} - {(\sqrt{r} e^{\frac{i θ}{2}})}^{2}}

= \int_{- \infty}^{+ \infty} \frac{d x}{(x - \sqrt{r} e^{\frac{i θ}{2}}) (x + \sqrt{r} e^{\frac{i θ}{2}})} = \frac{1}{2 \sqrt{r} e^{\frac{i θ}{2}}} \int_{- \infty}^{+ \infty} {\frac{1}{x - \sqrt{r} e^{\frac{i θ}{2}}} - \frac{1}{x + \sqrt{r} e^{\frac{i θ}{2}}}} d x

= \frac{1}{2 \sqrt{r} e^{\frac{i θ}{2}}} {\int_{- \infty}^{+ \infty} \frac{d x}{x - \sqrt{r} e^{\frac{i θ}{2}}} - \int_{- \infty}^{+ \infty} \frac{d x}{x - (- \sqrt{r} e^{\frac{i θ}{2}})}}

w e h a v e p r o v e d t h a t f o r a \in C \int_{- \infty}^{+ \infty} \frac{d x}{x - a} = i π i f I m (a) > 0 a n d

- i π i f I m (a) < 0 s o i f θ > 0 \Rightarrow

I = \frac{1}{2 \sqrt{r} e^{\frac{i θ}{2}}} {i π - (- i π)} = \frac{2 i π}{2 \sqrt{r} e^{\frac{i θ}{2}}} = \frac{i π}{\sqrt{r} e^{\frac{i θ}{2}}}

i f θ < 0 w e g e t I = \frac{1}{2 \sqrt{r} e^{\frac{i θ}{2}}} {- i π - (i π)} = - \frac{i π}{\sqrt{r} e^{\frac{i θ}{2}}}