let-f-x-0-dt-t-2-x-2-3-with-x-gt-0-1-find-a-explicit-form-off-x-1-calculate-0-dx-t-2-3-3-and-0-dt-t-2-4-3-2-find-the-value-of-A-0

Question Number 57899 by maxmathsup by imad last updated on 13/Apr/19

l e t f (x) = \int_{0}^{+ \infty} \frac{d t}{{(t^{2} + x^{2})}^{3}} w i t h x > 0

1) f i n d a e x p l i c i t f o r m o f f (x)

1) c a l c u l a t e \int_{0}^{\infty} \frac{d x}{{(t^{2} + 3)}^{3}} a n d \int_{0}^{\infty} \frac{d t}{{(t^{2} + 4)}^{3}}

2) f i n d t h e v a l u e o f A (θ) = \int_{0}^{\infty} \frac{d t}{{(t^{2} + {s i n}^{2} θ)}^{3}} w i t h 0 < θ < π .

Commented by maxmathsup by imad last updated on 17/Apr/19

1) w e h a v e f (x) = \frac{1}{2} \int_{- \infty}^{+ \infty} \frac{d t}{{(t^{2} + x^{2})}^{3}} l e t c o n s i d e r t h e c o m p l e x f u n c t i o n

φ (z) = \frac{1}{{(z^{2} + x^{2})}^{3}} p o l e s o f φ ?

w e h a v e φ (z) = \frac{1}{{(z - i x)}^{3} {(z + i x)}^{3}} s o t h e p o l e s o f a r e \bar{+} i x (t r i p l e s) \Rightarrow

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i x)

R e s (φ, i x) = {l i m}_{z \to i x} \frac{1}{(3 - 1)!} {{(z - i x)}^{3} φ (z)}^{(2)}

= {l i m}_{z \to i x} \frac{1}{2} {\frac{1}{{(z + i x)}^{3}}}^{(2)} w e h a v e

{\frac{1}{{(z + i x)}^{3}}}^{(1)} = - \frac{3 {(z + i x)}^{2}}{{(z + i x)}^{6}} = - \frac{3}{{(z + i x)}^{4}} \Rightarrow {\frac{1}{{(z + i x)}^{3}}}^{(2)}

= - 3 \frac{- 4 {(z + i x)}^{3}}{{(z + i x)}^{8}} = \frac{12}{{(z + i x)}^{5}} \Rightarrow R e s (φ, i x) = {l i m}_{z \to i x} \frac{6}{{(z + i x)}^{5}}

= \frac{6}{{(2 i x)}^{5}} = \frac{6}{2^{5} x^{5} i} = \frac{6}{32 {i x}^{5}} = \frac{3}{16 {i x}^{5}} \Rightarrow \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{3}{16 x^{5} i} = \frac{3 π}{8 x^{5}} \Rightarrow

f (x) = \frac{3 π}{16 x^{5}} (w i t h x > 0)

2) \int_{0}^{\infty} \frac{d t}{{(t^{2} + 3)}^{3}} = f (\sqrt{3}) = \frac{3 π}{16 {(\sqrt{3})}^{5}} = \frac{3 π}{16 {(\sqrt{3})}^{4} \sqrt{3}} = \frac{3 π}{9 \times 16 \sqrt{3}}

\int_{0}^{\infty} \frac{d t}{{(t^{2} + 4)}^{3}} = f (2) = \frac{3 π}{16 {(2)}^{5}} = \frac{3 π}{16 \times 32}

Commented by maxmathsup by imad last updated on 17/Apr/19

3) \int_{0}^{\infty} \frac{d t}{{(t^{2} + {s i n}^{2} θ)}^{3}} = f (s i n θ) = \frac{3 π}{16 {s i n}^{5} θ}