can knowing an improper integral dependence on first parameter help in studying dependence from second...
$begingroup$
I am interested in studying the dependence on parameter $a$ of integrals of this type
$$
int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx
$$
whereby real $k> 0$ and $a>0$ , while about real $f(x,k)$ we only know that it is continuous and differentiable, and that the above integral exists.
Let us suppose that through other means we also know the expression for
$$
frac{d}{dk} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;; int_{-infty} ^{infty} frac{d f(x,k)}{dk} , frac{1}{a^2+x^2}dx ;;=;;g(a,k)
$$
Could that provide information also about
$$
frac{d}{da} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;;-2a , int_{-infty} ^{infty} f(x,k); frac{1}{(a^2+x^2)^2}dx ;;;?
$$
at least in terms of bounds or inequalities ?
Mentally picturing a generic $f(x,k)$ somehow alternating between pieces of positive and negative values, while trying to naively visualise how positive and negative areas are affected in $a$ following the division by $(a^2+x^2)$, the first impression is that there must be some sort of connection between the two derivatives. I imagine that this kind of problems, or similar ones, may have already been systematically studied. I would much appreciate any suggestion about any relevant literature which may be of help.
integration inequality improper-integrals
asked Jan 21 at 12:49
LucaLuca
14
$endgroup$
add a comment |
$begingroup$
I am interested in studying the dependence on parameter $a$ of integrals of this type
$$
int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx
$$
whereby real $k> 0$ and $a>0$ , while about real $f(x,k)$ we only know that it is continuous and differentiable, and that the above integral exists.
Let us suppose that through other means we also know the expression for
$$
frac{d}{dk} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;; int_{-infty} ^{infty} frac{d f(x,k)}{dk} , frac{1}{a^2+x^2}dx ;;=;;g(a,k)
$$
Could that provide information also about
$$
frac{d}{da} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;;-2a , int_{-infty} ^{infty} f(x,k); frac{1}{(a^2+x^2)^2}dx ;;;?
$$
at least in terms of bounds or inequalities ?
Mentally picturing a generic $f(x,k)$ somehow alternating between pieces of positive and negative values, while trying to naively visualise how positive and negative areas are affected in $a$ following the division by $(a^2+x^2)$, the first impression is that there must be some sort of connection between the two derivatives. I imagine that this kind of problems, or similar ones, may have already been systematically studied. I would much appreciate any suggestion about any relevant literature which may be of help.
integration inequality improper-integrals
asked Jan 21 at 12:49
LucaLuca
14
$endgroup$
add a comment |
$begingroup$
I am interested in studying the dependence on parameter $a$ of integrals of this type
$$
int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx
$$
whereby real $k> 0$ and $a>0$ , while about real $f(x,k)$ we only know that it is continuous and differentiable, and that the above integral exists.
Let us suppose that through other means we also know the expression for
$$
frac{d}{dk} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;; int_{-infty} ^{infty} frac{d f(x,k)}{dk} , frac{1}{a^2+x^2}dx ;;=;;g(a,k)
$$
Could that provide information also about
$$
frac{d}{da} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;;-2a , int_{-infty} ^{infty} f(x,k); frac{1}{(a^2+x^2)^2}dx ;;;?
$$
at least in terms of bounds or inequalities ?
Mentally picturing a generic $f(x,k)$ somehow alternating between pieces of positive and negative values, while trying to naively visualise how positive and negative areas are affected in $a$ following the division by $(a^2+x^2)$, the first impression is that there must be some sort of connection between the two derivatives. I imagine that this kind of problems, or similar ones, may have already been systematically studied. I would much appreciate any suggestion about any relevant literature which may be of help.
integration inequality improper-integrals
asked Jan 21 at 12:49
LucaLuca
14
$endgroup$
I am interested in studying the dependence on parameter $a$ of integrals of this type
$$
int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx
$$
whereby real $k> 0$ and $a>0$ , while about real $f(x,k)$ we only know that it is continuous and differentiable, and that the above integral exists.
Let us suppose that through other means we also know the expression for
$$
frac{d}{dk} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;; int_{-infty} ^{infty} frac{d f(x,k)}{dk} , frac{1}{a^2+x^2}dx ;;=;;g(a,k)
$$
Could that provide information also about
$$
frac{d}{da} , int_{-infty} ^{infty} frac{f(x,k)}{a^2+x^2}dx ;;=;;-2a , int_{-infty} ^{infty} f(x,k); frac{1}{(a^2+x^2)^2}dx ;;;?
$$
at least in terms of bounds or inequalities ?
Mentally picturing a generic $f(x,k)$ somehow alternating between pieces of positive and negative values, while trying to naively visualise how positive and negative areas are affected in $a$ following the division by $(a^2+x^2)$, the first impression is that there must be some sort of connection between the two derivatives. I imagine that this kind of problems, or similar ones, may have already been systematically studied. I would much appreciate any suggestion about any relevant literature which may be of help.
integration inequality improper-integrals
integration inequality improper-integrals
asked Jan 21 at 12:49
LucaLuca
14
asked Jan 21 at 12:49
LucaLuca
14
edited Jan 22 at 10:12
Luca
asked Jan 21 at 12:49
LucaLuca
14
asked Jan 21 at 12:49
LucaLuca
14
asked Jan 21 at 12:49
LucaLuca
14
14
add a comment |
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