Integral involving Heaviside function in the exponential
up vote
0
down vote
favorite
I need to compute the following integral in the real domain that involves the Heaviside step function $Theta(x)$:
$$
int_0^{infty} e^{- alpha_0Theta(x)- alpha_1 x} dx
$$
with $alpha_0, alpha_1$ two real constants.
Is there some useful representation of the Heaviside $Theta$ function that I can exploit to calculate the integral?
Or should I compute the integral using some complex plane technique?
calculus integration
asked Nov 28 at 14:54
linello
17011
add a comment |
up vote
0
down vote
favorite
I need to compute the following integral in the real domain that involves the Heaviside step function $Theta(x)$:
$$
int_0^{infty} e^{- alpha_0Theta(x)- alpha_1 x} dx
$$
with $alpha_0, alpha_1$ two real constants.
Is there some useful representation of the Heaviside $Theta$ function that I can exploit to calculate the integral?
Or should I compute the integral using some complex plane technique?
calculus integration
asked Nov 28 at 14:54
linello
17011
1
Well, the step function is $+1$ on your entire domain of integration, so this is just $e^{-alpha_0} int_0^{infty} e^{-alpha_1 x} , dx$....
– T. Bongers
Nov 28 at 14:55
Yes, use the definition of that function.
– Yves Daoust
Nov 28 at 14:57
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need to compute the following integral in the real domain that involves the Heaviside step function $Theta(x)$:
$$
int_0^{infty} e^{- alpha_0Theta(x)- alpha_1 x} dx
$$
with $alpha_0, alpha_1$ two real constants.
Is there some useful representation of the Heaviside $Theta$ function that I can exploit to calculate the integral?
Or should I compute the integral using some complex plane technique?
calculus integration
asked Nov 28 at 14:54
linello
17011
I need to compute the following integral in the real domain that involves the Heaviside step function $Theta(x)$:
$$
int_0^{infty} e^{- alpha_0Theta(x)- alpha_1 x} dx
$$
with $alpha_0, alpha_1$ two real constants.
Is there some useful representation of the Heaviside $Theta$ function that I can exploit to calculate the integral?
Or should I compute the integral using some complex plane technique?
calculus integration
calculus integration
asked Nov 28 at 14:54
linello
17011
asked Nov 28 at 14:54
linello
17011
asked Nov 28 at 14:54
linello
17011
asked Nov 28 at 14:54
linello
17011
asked Nov 28 at 14:54
linello
17011
17011
1
Well, the step function is $+1$ on your entire domain of integration, so this is just $e^{-alpha_0} int_0^{infty} e^{-alpha_1 x} , dx$....
– T. Bongers
Nov 28 at 14:55
Yes, use the definition of that function.
– Yves Daoust
Nov 28 at 14:57
add a comment |
1
Well, the step function is $+1$ on your entire domain of integration, so this is just $e^{-alpha_0} int_0^{infty} e^{-alpha_1 x} , dx$....
– T. Bongers
Nov 28 at 14:55
Yes, use the definition of that function.
– Yves Daoust
Nov 28 at 14:57
1
1
Well, the step function is $+1$ on your entire domain of integration, so this is just $e^{-alpha_0} int_0^{infty} e^{-alpha_1 x} , dx$....
– T. Bongers
Nov 28 at 14:55
Well, the step function is $+1$ on your entire domain of integration, so this is just $e^{-alpha_0} int_0^{infty} e^{-alpha_1 x} , dx$....
– T. Bongers
Nov 28 at 14:55
Yes, use the definition of that function.
– Yves Daoust
Nov 28 at 14:57
Yes, use the definition of that function.
– Yves Daoust
Nov 28 at 14:57
add a comment |
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',
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%2f3017241%2fintegral-involving-heaviside-function-in-the-exponential%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3017241%2fintegral-involving-heaviside-function-in-the-exponential%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
1
Well, the step function is $+1$ on your entire domain of integration, so this is just $e^{-alpha_0} int_0^{infty} e^{-alpha_1 x} , dx$....
– T. Bongers
Nov 28 at 14:55
Yes, use the definition of that function.
– Yves Daoust
Nov 28 at 14:57