Limit $lim_{ttoinfty}int_{[1,+infty)}e^{-tx}frac{sin x}{x} dlambda(x)$
$f(t)=int_{[1,+infty)}e^{-tx}frac{sin x}{x} dlambda(x)$. $tin [1,+infty)$
How can I argue that $lim_{ttoinfty}f(t)=int_{[1,+infty)}lim_{ttoinfty}e^{-tx}frac{sin x}{x} dlambda(x)$? (switching limit and integral)
I know that $|e^{-tx}frac{sin x}{x}|le e^{-x}$ for $tge1$ and $int_1^infty e^{-x}<infty$ so I thought about using dominated convergence theorem dct
but what is my sequence $f_n$ in this case?
real-analysis integration analysis
add a comment |
$f(t)=int_{[1,+infty)}e^{-tx}frac{sin x}{x} dlambda(x)$. $tin [1,+infty)$
How can I argue that $lim_{ttoinfty}f(t)=int_{[1,+infty)}lim_{ttoinfty}e^{-tx}frac{sin x}{x} dlambda(x)$? (switching limit and integral)
I know that $|e^{-tx}frac{sin x}{x}|le e^{-x}$ for $tge1$ and $int_1^infty e^{-x}<infty$ so I thought about using dominated convergence theorem dct
but what is my sequence $f_n$ in this case?
real-analysis integration analysis
In the notation of wikipedia we have $g=e^{-x},f=e^{-tx}frac{sin x}{x}$ but what is my $f_n$?
– conrad
Nov 25 '18 at 10:56
Not sure I fully understand where the problem is... You know that, for every $x$, $$left|frac{sin x}{x}right|leqslant 1$$ hence $$|f(t)|leqslantint_1^inftyleft|e^{-tx}frac{sin x}{x}right|dxleqslantint_1^infty e^{-tx}dxleqslantint_0^infty e^{-tx}dx=frac1tto0$$
– Did
Nov 25 '18 at 12:05
add a comment |
$f(t)=int_{[1,+infty)}e^{-tx}frac{sin x}{x} dlambda(x)$. $tin [1,+infty)$
How can I argue that $lim_{ttoinfty}f(t)=int_{[1,+infty)}lim_{ttoinfty}e^{-tx}frac{sin x}{x} dlambda(x)$? (switching limit and integral)
I know that $|e^{-tx}frac{sin x}{x}|le e^{-x}$ for $tge1$ and $int_1^infty e^{-x}<infty$ so I thought about using dominated convergence theorem dct
but what is my sequence $f_n$ in this case?
real-analysis integration analysis
$f(t)=int_{[1,+infty)}e^{-tx}frac{sin x}{x} dlambda(x)$. $tin [1,+infty)$
How can I argue that $lim_{ttoinfty}f(t)=int_{[1,+infty)}lim_{ttoinfty}e^{-tx}frac{sin x}{x} dlambda(x)$? (switching limit and integral)
I know that $|e^{-tx}frac{sin x}{x}|le e^{-x}$ for $tge1$ and $int_1^infty e^{-x}<infty$ so I thought about using dominated convergence theorem dct
but what is my sequence $f_n$ in this case?
real-analysis integration analysis
real-analysis integration analysis
edited Nov 25 '18 at 12:18
Martin Sleziak
44.7k7115270
44.7k7115270
asked Nov 25 '18 at 10:40
conrad
757
757
In the notation of wikipedia we have $g=e^{-x},f=e^{-tx}frac{sin x}{x}$ but what is my $f_n$?
– conrad
Nov 25 '18 at 10:56
Not sure I fully understand where the problem is... You know that, for every $x$, $$left|frac{sin x}{x}right|leqslant 1$$ hence $$|f(t)|leqslantint_1^inftyleft|e^{-tx}frac{sin x}{x}right|dxleqslantint_1^infty e^{-tx}dxleqslantint_0^infty e^{-tx}dx=frac1tto0$$
– Did
Nov 25 '18 at 12:05
add a comment |
In the notation of wikipedia we have $g=e^{-x},f=e^{-tx}frac{sin x}{x}$ but what is my $f_n$?
– conrad
Nov 25 '18 at 10:56
Not sure I fully understand where the problem is... You know that, for every $x$, $$left|frac{sin x}{x}right|leqslant 1$$ hence $$|f(t)|leqslantint_1^inftyleft|e^{-tx}frac{sin x}{x}right|dxleqslantint_1^infty e^{-tx}dxleqslantint_0^infty e^{-tx}dx=frac1tto0$$
– Did
Nov 25 '18 at 12:05
In the notation of wikipedia we have $g=e^{-x},f=e^{-tx}frac{sin x}{x}$ but what is my $f_n$?
– conrad
Nov 25 '18 at 10:56
In the notation of wikipedia we have $g=e^{-x},f=e^{-tx}frac{sin x}{x}$ but what is my $f_n$?
– conrad
Nov 25 '18 at 10:56
Not sure I fully understand where the problem is... You know that, for every $x$, $$left|frac{sin x}{x}right|leqslant 1$$ hence $$|f(t)|leqslantint_1^inftyleft|e^{-tx}frac{sin x}{x}right|dxleqslantint_1^infty e^{-tx}dxleqslantint_0^infty e^{-tx}dx=frac1tto0$$
– Did
Nov 25 '18 at 12:05
Not sure I fully understand where the problem is... You know that, for every $x$, $$left|frac{sin x}{x}right|leqslant 1$$ hence $$|f(t)|leqslantint_1^inftyleft|e^{-tx}frac{sin x}{x}right|dxleqslantint_1^infty e^{-tx}dxleqslantint_0^infty e^{-tx}dx=frac1tto0$$
– Did
Nov 25 '18 at 12:05
add a comment |
1 Answer
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Theorem:
A functional limit $lim_{xto a}f(x)$ converges to some limit $L$ if and only if for every sequence ${x_n}_{ninBbb N}$ such that $x_nneq a$ for all $ninBbb N$ and that converges to $a$ the sequence ${f(x_n)}_{ninBbb N}$ converges to $L$.
Hence
$$lim_{ttoinfty}int_X f(x,t), dx=Liff lim_{ntoinfty}int_X f(x,t_n), dx=Ltag1$$
for every sequence ${t_n}_{ninBbb N}$ that converges to infinity. If we set $f_n(x):=f(x,t_n)$ then we have the equivalent expression
$$lim_{ttoinfty}int_X f(x,t), dx=Lifflim_{ntoinfty}int_X f_n(x), dx=Ltag2$$
for every sequence ${t_n}_{ninBbb N}toinfty$. Now using the dominated convergence theorem it is easy to check that the RHS of $(2)$ converges to $L=0$ for any chosen sequence ${t_n}_{ninBbb N}toinfty$.
add a comment |
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1 Answer
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1 Answer
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Theorem:
A functional limit $lim_{xto a}f(x)$ converges to some limit $L$ if and only if for every sequence ${x_n}_{ninBbb N}$ such that $x_nneq a$ for all $ninBbb N$ and that converges to $a$ the sequence ${f(x_n)}_{ninBbb N}$ converges to $L$.
Hence
$$lim_{ttoinfty}int_X f(x,t), dx=Liff lim_{ntoinfty}int_X f(x,t_n), dx=Ltag1$$
for every sequence ${t_n}_{ninBbb N}$ that converges to infinity. If we set $f_n(x):=f(x,t_n)$ then we have the equivalent expression
$$lim_{ttoinfty}int_X f(x,t), dx=Lifflim_{ntoinfty}int_X f_n(x), dx=Ltag2$$
for every sequence ${t_n}_{ninBbb N}toinfty$. Now using the dominated convergence theorem it is easy to check that the RHS of $(2)$ converges to $L=0$ for any chosen sequence ${t_n}_{ninBbb N}toinfty$.
add a comment |
Theorem:
A functional limit $lim_{xto a}f(x)$ converges to some limit $L$ if and only if for every sequence ${x_n}_{ninBbb N}$ such that $x_nneq a$ for all $ninBbb N$ and that converges to $a$ the sequence ${f(x_n)}_{ninBbb N}$ converges to $L$.
Hence
$$lim_{ttoinfty}int_X f(x,t), dx=Liff lim_{ntoinfty}int_X f(x,t_n), dx=Ltag1$$
for every sequence ${t_n}_{ninBbb N}$ that converges to infinity. If we set $f_n(x):=f(x,t_n)$ then we have the equivalent expression
$$lim_{ttoinfty}int_X f(x,t), dx=Lifflim_{ntoinfty}int_X f_n(x), dx=Ltag2$$
for every sequence ${t_n}_{ninBbb N}toinfty$. Now using the dominated convergence theorem it is easy to check that the RHS of $(2)$ converges to $L=0$ for any chosen sequence ${t_n}_{ninBbb N}toinfty$.
add a comment |
Theorem:
A functional limit $lim_{xto a}f(x)$ converges to some limit $L$ if and only if for every sequence ${x_n}_{ninBbb N}$ such that $x_nneq a$ for all $ninBbb N$ and that converges to $a$ the sequence ${f(x_n)}_{ninBbb N}$ converges to $L$.
Hence
$$lim_{ttoinfty}int_X f(x,t), dx=Liff lim_{ntoinfty}int_X f(x,t_n), dx=Ltag1$$
for every sequence ${t_n}_{ninBbb N}$ that converges to infinity. If we set $f_n(x):=f(x,t_n)$ then we have the equivalent expression
$$lim_{ttoinfty}int_X f(x,t), dx=Lifflim_{ntoinfty}int_X f_n(x), dx=Ltag2$$
for every sequence ${t_n}_{ninBbb N}toinfty$. Now using the dominated convergence theorem it is easy to check that the RHS of $(2)$ converges to $L=0$ for any chosen sequence ${t_n}_{ninBbb N}toinfty$.
Theorem:
A functional limit $lim_{xto a}f(x)$ converges to some limit $L$ if and only if for every sequence ${x_n}_{ninBbb N}$ such that $x_nneq a$ for all $ninBbb N$ and that converges to $a$ the sequence ${f(x_n)}_{ninBbb N}$ converges to $L$.
Hence
$$lim_{ttoinfty}int_X f(x,t), dx=Liff lim_{ntoinfty}int_X f(x,t_n), dx=Ltag1$$
for every sequence ${t_n}_{ninBbb N}$ that converges to infinity. If we set $f_n(x):=f(x,t_n)$ then we have the equivalent expression
$$lim_{ttoinfty}int_X f(x,t), dx=Lifflim_{ntoinfty}int_X f_n(x), dx=Ltag2$$
for every sequence ${t_n}_{ninBbb N}toinfty$. Now using the dominated convergence theorem it is easy to check that the RHS of $(2)$ converges to $L=0$ for any chosen sequence ${t_n}_{ninBbb N}toinfty$.
edited Nov 25 '18 at 12:16
answered Nov 25 '18 at 12:01
Masacroso
12.9k41746
12.9k41746
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In the notation of wikipedia we have $g=e^{-x},f=e^{-tx}frac{sin x}{x}$ but what is my $f_n$?
– conrad
Nov 25 '18 at 10:56
Not sure I fully understand where the problem is... You know that, for every $x$, $$left|frac{sin x}{x}right|leqslant 1$$ hence $$|f(t)|leqslantint_1^inftyleft|e^{-tx}frac{sin x}{x}right|dxleqslantint_1^infty e^{-tx}dxleqslantint_0^infty e^{-tx}dx=frac1tto0$$
– Did
Nov 25 '18 at 12:05