What does an integral with only one integration limit mean?
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I am supervising an end of degree project related to Sturm-Liouville problems. In the paper Singular Sturm comparison theorems I saw the next strange integrals in Theorem 1:
Let $P(x)$, $p(x)$ be continuous functions on the open, finite or infinite interval $(a,b)$ (but not necessarily at its endpoints), and $P(x)geq p(x)$, $P (x)notequiv p(x)$ on $(a, b)$.
(i) Suppose that the differential equation
$$u′′+p(x)u=0,quad a<x<b,$$
has a solution $u$ which satisfies the boundary conditions
$$int_afrac{dx}{u^2(x)}=infty, int^bfrac{dx}{u^2(x)}=infty.$$
In the paper, there is no explanation about how they are defined.
There is one reference in the paper to the book Ordinary Differential Equations by Philip Hartman. In this book, the notation is used extensively but I could neither found their definition.
My thoughts: they don't seem to be primitives; they neither seem to be improper integrals in an infinite interval.
Can someone help me, please?
notation
asked Nov 27 at 7:56
Math wind
164
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up vote
3
down vote
favorite
I am supervising an end of degree project related to Sturm-Liouville problems. In the paper Singular Sturm comparison theorems I saw the next strange integrals in Theorem 1:
Let $P(x)$, $p(x)$ be continuous functions on the open, finite or infinite interval $(a,b)$ (but not necessarily at its endpoints), and $P(x)geq p(x)$, $P (x)notequiv p(x)$ on $(a, b)$.
(i) Suppose that the differential equation
$$u′′+p(x)u=0,quad a<x<b,$$
has a solution $u$ which satisfies the boundary conditions
$$int_afrac{dx}{u^2(x)}=infty, int^bfrac{dx}{u^2(x)}=infty.$$
In the paper, there is no explanation about how they are defined.
There is one reference in the paper to the book Ordinary Differential Equations by Philip Hartman. In this book, the notation is used extensively but I could neither found their definition.
My thoughts: they don't seem to be primitives; they neither seem to be improper integrals in an infinite interval.
Can someone help me, please?
notation
asked Nov 27 at 7:56
Math wind
164
This is a just a guess, but the notation suggests the following interpretation to me $$int_a f(x)dx=lim_{cto a+}int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context?
– saulspatz
Nov 27 at 8:13
2
IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant.
– Yves Daoust
Nov 27 at 10:06
@YvesDaoust, thank you very much for your answer. I think that this make sense.
– Math wind
Nov 28 at 7:54
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I am supervising an end of degree project related to Sturm-Liouville problems. In the paper Singular Sturm comparison theorems I saw the next strange integrals in Theorem 1:
Let $P(x)$, $p(x)$ be continuous functions on the open, finite or infinite interval $(a,b)$ (but not necessarily at its endpoints), and $P(x)geq p(x)$, $P (x)notequiv p(x)$ on $(a, b)$.
(i) Suppose that the differential equation
$$u′′+p(x)u=0,quad a<x<b,$$
has a solution $u$ which satisfies the boundary conditions
$$int_afrac{dx}{u^2(x)}=infty, int^bfrac{dx}{u^2(x)}=infty.$$
In the paper, there is no explanation about how they are defined.
There is one reference in the paper to the book Ordinary Differential Equations by Philip Hartman. In this book, the notation is used extensively but I could neither found their definition.
My thoughts: they don't seem to be primitives; they neither seem to be improper integrals in an infinite interval.
Can someone help me, please?
notation
asked Nov 27 at 7:56
Math wind
164
I am supervising an end of degree project related to Sturm-Liouville problems. In the paper Singular Sturm comparison theorems I saw the next strange integrals in Theorem 1:
Let $P(x)$, $p(x)$ be continuous functions on the open, finite or infinite interval $(a,b)$ (but not necessarily at its endpoints), and $P(x)geq p(x)$, $P (x)notequiv p(x)$ on $(a, b)$.
(i) Suppose that the differential equation
$$u′′+p(x)u=0,quad a<x<b,$$
has a solution $u$ which satisfies the boundary conditions
$$int_afrac{dx}{u^2(x)}=infty, int^bfrac{dx}{u^2(x)}=infty.$$
In the paper, there is no explanation about how they are defined.
There is one reference in the paper to the book Ordinary Differential Equations by Philip Hartman. In this book, the notation is used extensively but I could neither found their definition.
My thoughts: they don't seem to be primitives; they neither seem to be improper integrals in an infinite interval.
Can someone help me, please?
notation
notation
asked Nov 27 at 7:56
Math wind
164
asked Nov 27 at 7:56
Math wind
164
edited Nov 27 at 12:40
asked Nov 27 at 7:56
Math wind
164
asked Nov 27 at 7:56
Math wind
164
asked Nov 27 at 7:56
Math wind
164
164
This is a just a guess, but the notation suggests the following interpretation to me $$int_a f(x)dx=lim_{cto a+}int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context?
– saulspatz
Nov 27 at 8:13
2
IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant.
– Yves Daoust
Nov 27 at 10:06
@YvesDaoust, thank you very much for your answer. I think that this make sense.
– Math wind
Nov 28 at 7:54
add a comment |
This is a just a guess, but the notation suggests the following interpretation to me $$int_a f(x)dx=lim_{cto a+}int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context?
– saulspatz
Nov 27 at 8:13
2
IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant.
– Yves Daoust
Nov 27 at 10:06
@YvesDaoust, thank you very much for your answer. I think that this make sense.
– Math wind
Nov 28 at 7:54
This is a just a guess, but the notation suggests the following interpretation to me $$int_a f(x)dx=lim_{cto a+}int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context?
– saulspatz
Nov 27 at 8:13
This is a just a guess, but the notation suggests the following interpretation to me $$int_a f(x)dx=lim_{cto a+}int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context?
– saulspatz
Nov 27 at 8:13
2
2
IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant.
– Yves Daoust
Nov 27 at 10:06
IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant.
– Yves Daoust
Nov 27 at 10:06
@YvesDaoust, thank you very much for your answer. I think that this make sense.
– Math wind
Nov 28 at 7:54
@YvesDaoust, thank you very much for your answer. I think that this make sense.
– Math wind
Nov 28 at 7:54
add a comment |
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This is a just a guess, but the notation suggests the following interpretation to me $$int_a f(x)dx=lim_{cto a+}int_c^d f(x)dx,$$ and similarly for the other. Does this make sense in context?
– saulspatz
Nov 27 at 8:13
2
IMO, "near $x=a$" and "near $x=b$". As the integral diverges at $a$ and $b$, the other bound is unimportant.
– Yves Daoust
Nov 27 at 10:06
@YvesDaoust, thank you very much for your answer. I think that this make sense.
– Math wind
Nov 28 at 7:54