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

Question Number 67851 by mathmax by abdo last updated on 01/Sep/19

f i n d \int \frac{d x}{x^{2} - z} w i t h z f r o m C .

Commented by MJS last updated on 01/Sep/19

{what}^{'} s the problem / mistake if we just

calculate it as if z \in R ?

\int \frac{d x}{x^{2} - z} = \frac{1}{2 \sqrt{z}} \ln \frac{x - \sqrt{z}}{x + \sqrt{z}} + C

for \underset{0}{\int^{\infty}} \frac{d x}{x^{2} - z} this would give - \frac{π}{2 \sqrt{z}} i

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

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

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

= \frac{e^{- \frac{i θ}{2}}}{2 \sqrt{r}} l n (\frac{x - \sqrt{r} e^{\frac{i θ}{2}}}{x + \sqrt{r} e^{\frac{i θ}{2}}}) + C .

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

s i r m j s y o u r a n s w e r i s c o r r e c t .

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

\int_{0}^{\infty} \frac{d x}{x^{2} - z} = \frac{e^{- \frac{i θ}{2}}}{2 \sqrt{r}} {[l n (\frac{x - \sqrt{r} e^{\frac{i θ}{2}}}{x + \sqrt{r} e^{\frac{i θ}{2}}})]}_{0}^{+ \infty} = \frac{e^{- \frac{i θ}{2}}}{2 \sqrt{r}} (- l n (- 1)) = \frac{i π}{2 (2 \sqrt{r}) e^{\frac{i θ}{2}}}

= \frac{i π}{4 \sqrt{r} e^{\frac{i θ}{2}}} w i t h z = r e^{i θ} .

Commented by MJS last updated on 01/Sep/19

thank you

I just {wasn}^{'} t sure if this was allowed in C

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