Mixing
Fourier transform of $f(x) = frac {1}{4pi x^2 + k^2}$ in $Bbb R^3$
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I got a function $f(x) = frac {1}{4pi x^2 + k^2}$, where $k gt 0$ is a constant. And I wanted do make the fourier transform in $Bbb R^3$ (it is solved in my textbook). But in the solution it was said that the transform is in the sense of the distributions. But why? Why this function isn't Lebesgue integrable, in $L^1(Bbb R^3)$?
I understand that $frac {1}{x^2}$ isn't in $L^1$, because the function isn't continuous in $x=0$. Even though $frac {1}{x^2}$ goes to zero for $x to infty$. But I don't see any discontinuity in $frac {1}{4pi x^2 + k^2}$. So why it is needed to use the radial distribution. Because of the dimension? I am confused.
calculus lebesgue-integral fourier-transform
asked Jan 26 at 14:35
LeifLeif
611314
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add a comment |
$begingroup$
I got a function $f(x) = frac {1}{4pi x^2 + k^2}$, where $k gt 0$ is a constant. And I wanted do make the fourier transform in $Bbb R^3$ (it is solved in my textbook). But in the solution it was said that the transform is in the sense of the distributions. But why? Why this function isn't Lebesgue integrable, in $L^1(Bbb R^3)$?
I understand that $frac {1}{x^2}$ isn't in $L^1$, because the function isn't continuous in $x=0$. Even though $frac {1}{x^2}$ goes to zero for $x to infty$. But I don't see any discontinuity in $frac {1}{4pi x^2 + k^2}$. So why it is needed to use the radial distribution. Because of the dimension? I am confused.
calculus lebesgue-integral fourier-transform
asked Jan 26 at 14:35
LeifLeif
611314
$endgroup$
$begingroup$
Is $x$ a radial coordinate of a spherically symmetric function, or is $x$ just the normal cartesian $x$ coordinate?
$endgroup$
– Andy Walls
Jan 27 at 0:10
$begingroup$
For a 3-D Fourier Transform of a function with radial symmetry, you may want to look at this answer to a similar question: math.stackexchange.com/a/3029986/441161
$endgroup$
– Andy Walls
Jan 27 at 0:13
$begingroup$
Thank you. Fun Fact: the similar question was posted by my classmate.
$endgroup$
– Leif
Jan 27 at 3:07
add a comment |
$begingroup$
I got a function $f(x) = frac {1}{4pi x^2 + k^2}$, where $k gt 0$ is a constant. And I wanted do make the fourier transform in $Bbb R^3$ (it is solved in my textbook). But in the solution it was said that the transform is in the sense of the distributions. But why? Why this function isn't Lebesgue integrable, in $L^1(Bbb R^3)$?
I understand that $frac {1}{x^2}$ isn't in $L^1$, because the function isn't continuous in $x=0$. Even though $frac {1}{x^2}$ goes to zero for $x to infty$. But I don't see any discontinuity in $frac {1}{4pi x^2 + k^2}$. So why it is needed to use the radial distribution. Because of the dimension? I am confused.
calculus lebesgue-integral fourier-transform
asked Jan 26 at 14:35
LeifLeif
611314
$endgroup$
I got a function $f(x) = frac {1}{4pi x^2 + k^2}$, where $k gt 0$ is a constant. And I wanted do make the fourier transform in $Bbb R^3$ (it is solved in my textbook). But in the solution it was said that the transform is in the sense of the distributions. But why? Why this function isn't Lebesgue integrable, in $L^1(Bbb R^3)$?
I understand that $frac {1}{x^2}$ isn't in $L^1$, because the function isn't continuous in $x=0$. Even though $frac {1}{x^2}$ goes to zero for $x to infty$. But I don't see any discontinuity in $frac {1}{4pi x^2 + k^2}$. So why it is needed to use the radial distribution. Because of the dimension? I am confused.
calculus lebesgue-integral fourier-transform
calculus lebesgue-integral fourier-transform
asked Jan 26 at 14:35
LeifLeif
611314
asked Jan 26 at 14:35
LeifLeif
611314
asked Jan 26 at 14:35
LeifLeif
611314
asked Jan 26 at 14:35
LeifLeif
611314
asked Jan 26 at 14:35
LeifLeif
611314
611314
$begingroup$
Is $x$ a radial coordinate of a spherically symmetric function, or is $x$ just the normal cartesian $x$ coordinate?
$endgroup$
– Andy Walls
Jan 27 at 0:10
$begingroup$
For a 3-D Fourier Transform of a function with radial symmetry, you may want to look at this answer to a similar question: math.stackexchange.com/a/3029986/441161
$endgroup$
– Andy Walls
Jan 27 at 0:13
$begingroup$
Thank you. Fun Fact: the similar question was posted by my classmate.
$endgroup$
– Leif
Jan 27 at 3:07
add a comment |
$begingroup$
Is $x$ a radial coordinate of a spherically symmetric function, or is $x$ just the normal cartesian $x$ coordinate?
$endgroup$
– Andy Walls
Jan 27 at 0:10
$begingroup$
For a 3-D Fourier Transform of a function with radial symmetry, you may want to look at this answer to a similar question: math.stackexchange.com/a/3029986/441161
$endgroup$
– Andy Walls
Jan 27 at 0:13
$begingroup$
Thank you. Fun Fact: the similar question was posted by my classmate.
$endgroup$
– Leif
Jan 27 at 3:07
$begingroup$
Is $x$ a radial coordinate of a spherically symmetric function, or is $x$ just the normal cartesian $x$ coordinate?
$endgroup$
– Andy Walls
Jan 27 at 0:10
$begingroup$
Is $x$ a radial coordinate of a spherically symmetric function, or is $x$ just the normal cartesian $x$ coordinate?
$endgroup$
– Andy Walls
Jan 27 at 0:10
$begingroup$
For a 3-D Fourier Transform of a function with radial symmetry, you may want to look at this answer to a similar question: math.stackexchange.com/a/3029986/441161
$endgroup$
– Andy Walls
Jan 27 at 0:13
$begingroup$
For a 3-D Fourier Transform of a function with radial symmetry, you may want to look at this answer to a similar question: math.stackexchange.com/a/3029986/441161
$endgroup$
– Andy Walls
Jan 27 at 0:13
$begingroup$
Thank you. Fun Fact: the similar question was posted by my classmate.
$endgroup$
– Leif
Jan 27 at 3:07
$begingroup$
Thank you. Fun Fact: the similar question was posted by my classmate.
$endgroup$
– Leif
Jan 27 at 3:07
add a comment |
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$begingroup$
Is $x$ a radial coordinate of a spherically symmetric function, or is $x$ just the normal cartesian $x$ coordinate?
$endgroup$
– Andy Walls
Jan 27 at 0:10
$begingroup$
For a 3-D Fourier Transform of a function with radial symmetry, you may want to look at this answer to a similar question: math.stackexchange.com/a/3029986/441161
$endgroup$
– Andy Walls
Jan 27 at 0:13
$begingroup$
Thank you. Fun Fact: the similar question was posted by my classmate.
$endgroup$
– Leif
Jan 27 at 3:07