Mixing
Extension of the initial value theorem
$begingroup$
I'm trying to prove this extension of the initial value theorem for Laplace transforms.
If
$$lim_{sto infty}s^{v+1}L{f(t)}=C$$
, with $L$ the laplace transform
then
$$D^{v}f(0)=C$$
I know the initial value theorem
$$lim_{sto infty}sF(s)=f(0)$$
So now i try to prove that
$$lim_{sto infty}s^{v+1}F(s)=D^vf(0).$$
I know that
$$ L{D^vf(t)}=s^vF(s)-sum_{k=0}^{m-1}s^{m-k-1}D^{k-m+v}f(0) $$
and i can prove that
$$lim_{stoinfty}L{D^vf(t)}=0.$$
Now it leaves me desperate to unite the knowledge and to find a prove.
Any thoughts on the problem?
complex-analysis laplace-transform initial-value-problems
asked Jan 8 at 21:00
A.SoeA.Soe
11
$endgroup$
add a comment |
$begingroup$
I'm trying to prove this extension of the initial value theorem for Laplace transforms.
If
$$lim_{sto infty}s^{v+1}L{f(t)}=C$$
, with $L$ the laplace transform
then
$$D^{v}f(0)=C$$
I know the initial value theorem
$$lim_{sto infty}sF(s)=f(0)$$
So now i try to prove that
$$lim_{sto infty}s^{v+1}F(s)=D^vf(0).$$
I know that
$$ L{D^vf(t)}=s^vF(s)-sum_{k=0}^{m-1}s^{m-k-1}D^{k-m+v}f(0) $$
and i can prove that
$$lim_{stoinfty}L{D^vf(t)}=0.$$
Now it leaves me desperate to unite the knowledge and to find a prove.
Any thoughts on the problem?
complex-analysis laplace-transform initial-value-problems
asked Jan 8 at 21:00
A.SoeA.Soe
11
$endgroup$
add a comment |
$begingroup$
I'm trying to prove this extension of the initial value theorem for Laplace transforms.
If
$$lim_{sto infty}s^{v+1}L{f(t)}=C$$
, with $L$ the laplace transform
then
$$D^{v}f(0)=C$$
I know the initial value theorem
$$lim_{sto infty}sF(s)=f(0)$$
So now i try to prove that
$$lim_{sto infty}s^{v+1}F(s)=D^vf(0).$$
I know that
$$ L{D^vf(t)}=s^vF(s)-sum_{k=0}^{m-1}s^{m-k-1}D^{k-m+v}f(0) $$
and i can prove that
$$lim_{stoinfty}L{D^vf(t)}=0.$$
Now it leaves me desperate to unite the knowledge and to find a prove.
Any thoughts on the problem?
complex-analysis laplace-transform initial-value-problems
asked Jan 8 at 21:00
A.SoeA.Soe
11
$endgroup$
I'm trying to prove this extension of the initial value theorem for Laplace transforms.
If
$$lim_{sto infty}s^{v+1}L{f(t)}=C$$
, with $L$ the laplace transform
then
$$D^{v}f(0)=C$$
I know the initial value theorem
$$lim_{sto infty}sF(s)=f(0)$$
So now i try to prove that
$$lim_{sto infty}s^{v+1}F(s)=D^vf(0).$$
I know that
$$ L{D^vf(t)}=s^vF(s)-sum_{k=0}^{m-1}s^{m-k-1}D^{k-m+v}f(0) $$
and i can prove that
$$lim_{stoinfty}L{D^vf(t)}=0.$$
Now it leaves me desperate to unite the knowledge and to find a prove.
Any thoughts on the problem?
complex-analysis laplace-transform initial-value-problems
complex-analysis laplace-transform initial-value-problems
asked Jan 8 at 21:00
A.SoeA.Soe
11
asked Jan 8 at 21:00
A.SoeA.Soe
11
asked Jan 8 at 21:00
A.SoeA.Soe
11
asked Jan 8 at 21:00
A.SoeA.Soe
11
asked Jan 8 at 21:00
A.SoeA.Soe
11
11
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066711%2fextension-of-the-initial-value-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066711%2fextension-of-the-initial-value-theorem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
