Mixing
Integrate $xe^{-bx/d}mathrm{erfc}(ax+c)$
$begingroup$
I want to calculate and evaluate the following integral:
$$frac{B}{2 D}int_{0}^{infty} xe^{frac{-Bx}{D}} erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})$$
My idea was to integrate by parts by setting: $$u= erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}}), du=-frac{1}{sqrt{Dt}} e^{-(frac{x+x_{0}-Bt}{2sqrt{Dt}})^2}$$
$$dv=xe^{-frac{Bx}{D}},v=e^{-frac{Bx}{D}}(frac{Bx}{D}x+1)*frac{D^2}{B^2}$$
I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated.
And of course, erfc is the complementary error function with:
$$erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})=frac{2}{sqrt{pi}}int_{frac{x+x_{0}-Bt}{2sqrt{Dt}}}^{infty} e^{-z^2} dz $$
real-analysis calculus integration numerical-methods special-functions
edited Jan 29 at 16:05
ablmf
2,58552452
asked Jan 29 at 16:03
George FarahGeorge Farah
65
$endgroup$
add a comment |
$begingroup$
I want to calculate and evaluate the following integral:
$$frac{B}{2 D}int_{0}^{infty} xe^{frac{-Bx}{D}} erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})$$
My idea was to integrate by parts by setting: $$u= erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}}), du=-frac{1}{sqrt{Dt}} e^{-(frac{x+x_{0}-Bt}{2sqrt{Dt}})^2}$$
$$dv=xe^{-frac{Bx}{D}},v=e^{-frac{Bx}{D}}(frac{Bx}{D}x+1)*frac{D^2}{B^2}$$
I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated.
And of course, erfc is the complementary error function with:
$$erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})=frac{2}{sqrt{pi}}int_{frac{x+x_{0}-Bt}{2sqrt{Dt}}}^{infty} e^{-z^2} dz $$
real-analysis calculus integration numerical-methods special-functions
edited Jan 29 at 16:05
ablmf
2,58552452
asked Jan 29 at 16:03
George FarahGeorge Farah
65
$endgroup$
$begingroup$
CAS like Mathematica can handle this integral.
$endgroup$
– Mariusz Iwaniuk
Jan 29 at 18:12
add a comment |
$begingroup$
I want to calculate and evaluate the following integral:
$$frac{B}{2 D}int_{0}^{infty} xe^{frac{-Bx}{D}} erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})$$
My idea was to integrate by parts by setting: $$u= erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}}), du=-frac{1}{sqrt{Dt}} e^{-(frac{x+x_{0}-Bt}{2sqrt{Dt}})^2}$$
$$dv=xe^{-frac{Bx}{D}},v=e^{-frac{Bx}{D}}(frac{Bx}{D}x+1)*frac{D^2}{B^2}$$
I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated.
And of course, erfc is the complementary error function with:
$$erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})=frac{2}{sqrt{pi}}int_{frac{x+x_{0}-Bt}{2sqrt{Dt}}}^{infty} e^{-z^2} dz $$
real-analysis calculus integration numerical-methods special-functions
edited Jan 29 at 16:05
ablmf
2,58552452
asked Jan 29 at 16:03
George FarahGeorge Farah
65
$endgroup$
I want to calculate and evaluate the following integral:
$$frac{B}{2 D}int_{0}^{infty} xe^{frac{-Bx}{D}} erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})$$
My idea was to integrate by parts by setting: $$u= erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}}), du=-frac{1}{sqrt{Dt}} e^{-(frac{x+x_{0}-Bt}{2sqrt{Dt}})^2}$$
$$dv=xe^{-frac{Bx}{D}},v=e^{-frac{Bx}{D}}(frac{Bx}{D}x+1)*frac{D^2}{B^2}$$
I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated.
And of course, erfc is the complementary error function with:
$$erfc(frac{x+x_{0}-Bt}{2sqrt{Dt}})=frac{2}{sqrt{pi}}int_{frac{x+x_{0}-Bt}{2sqrt{Dt}}}^{infty} e^{-z^2} dz $$
real-analysis calculus integration numerical-methods special-functions
real-analysis calculus integration numerical-methods special-functions
edited Jan 29 at 16:05
ablmf
2,58552452
asked Jan 29 at 16:03
George FarahGeorge Farah
65
edited Jan 29 at 16:05
ablmf
2,58552452
asked Jan 29 at 16:03
George FarahGeorge Farah
65
edited Jan 29 at 16:05
ablmf
2,58552452
edited Jan 29 at 16:05
ablmf
2,58552452
edited Jan 29 at 16:05
ablmf
2,58552452
2,58552452
asked Jan 29 at 16:03
George FarahGeorge Farah
65
asked Jan 29 at 16:03
George FarahGeorge Farah
65
asked Jan 29 at 16:03
George FarahGeorge Farah
65
65
$begingroup$
CAS like Mathematica can handle this integral.
$endgroup$
– Mariusz Iwaniuk
Jan 29 at 18:12
add a comment |
$begingroup$
CAS like Mathematica can handle this integral.
$endgroup$
– Mariusz Iwaniuk
Jan 29 at 18:12
$begingroup$
CAS like Mathematica can handle this integral.
$endgroup$
– Mariusz Iwaniuk
Jan 29 at 18:12
$begingroup$
CAS like Mathematica can handle this integral.
$endgroup$
– Mariusz Iwaniuk
Jan 29 at 18:12
add a comment |
0
active
oldest
votes
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%2f3092347%2fintegrate-xe-bx-d-mathrmerfcaxc%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%2f3092347%2fintegrate-xe-bx-d-mathrmerfcaxc%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
$begingroup$
CAS like Mathematica can handle this integral.
$endgroup$
– Mariusz Iwaniuk
Jan 29 at 18:12