laplace-transform-L-sin-t-

Question Number 205599 by mnjuly1970 last updated on 25/Mar/24

l a p l a c e t r a n s f o r m \dots

L {s i n (\sqrt{t})} = ?

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Commented by SANOGO last updated on 25/Mar/24

m e r c i b e a u c o u p

Answered by Berbere last updated on 25/Mar/24

L {s i n (\sqrt{t})} (x) = \int_{0}^{\infty} s i n (\sqrt{t}) e^{- x t} d t; x > 0

\sqrt{t} = y

= \int_{0}^{\infty} s i n (y) e^{- {x y}^{2}} 2 y d y \overset{I B P}{=} {[- \frac{e^{- {x y}^{2}}}{x} s i n (y)]}_{0}^{\infty} + \frac{1}{x} \int_{0}^{\infty} c o s (y) e^{- {x y}^{2}} d y

= \frac{1}{x} R e \int_{0}^{\infty} e^{- {x y}^{2} + i y} d y = \frac{1}{x} R e \int_{0}^{\infty} e^{- {(y \sqrt{x} + \frac{i}{2 \sqrt{x}})}^{2} - \frac{1}{4 x}} d y

y \sqrt{x} . = u, \frac{e^{- \frac{1}{4 x}}}{x \sqrt{x}} R e \int_{0}^{\infty} e^{- {(u + \frac{i}{2 \sqrt{x}})}^{2}} d u

X = u + \frac{i}{2 \sqrt{x}}; \frac{e^{- \frac{1}{4 x}}}{x \sqrt{x}} R e \int_{0 + \frac{i}{2 \sqrt{x}}}^{\infty + \frac{i}{2 \sqrt{x}}} e^{- X^{2}} d X;

l e t r e c t a n g l e o f D q (0, 0) (0, \frac{i}{2 \sqrt{x}}); (\infty, \frac{i}{2 \sqrt{x}}); (\infty, 0)

f (z) = e^{- z^{2}}

\int_{D} f (z) d z = 0 H o l o m o r p h i c f u n c t i o n c a u c h y T h e o r e m

\int_{0}^{\frac{i}{2 \sqrt{x}}} f (z) d z + \int_{\frac{i}{2 \sqrt{x}}}^{\infty + \frac{i}{2 \sqrt{x}}} f (z) d z + \int_{\infty + \frac{i}{2 \sqrt{x}}}^{\infty} f (z) {d z}_{= 0} + \int_{\infty}^{0} f (z) d z = 0

z = x + i y ∣ f (z) ∣⩽ e^{- y^{2}} . e^{- x^{2}} \to 0 x \to \infty

\int_{0}^{\infty} e^{- z^{2}} = \frac{1}{2} Γ (\frac{1}{2}) = \frac{\sqrt{π}}{2}; \int_{0}^{\frac{i}{2 \sqrt{x}}} e^{- z^{2}} d z; z = i t

= i \int_{0}^{\frac{1}{2 \sqrt{x}}} e^{t^{2}} d t i m a g i n a i r P u r

R e \int_{0 + \frac{i}{2 \sqrt{x}}}^{\infty + \frac{i}{2 \sqrt{x}}} e^{- X^{2}} = \frac{\sqrt{π}}{2}

L (s i n (\sqrt{t}) (x) = \frac{e^{- \frac{1}{4 x}}}{x \sqrt{x}} . \frac{\sqrt{π}}{2}

Commented by mnjuly1970 last updated on 25/Mar/24

t h a n k s a l o t M a s t e r

Commented by Berbere last updated on 27/Mar/24

w i t h e p l e a s u r

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