A problem when integrate Cos[n*x]*Cos[k*x]
$begingroup$
When integrate the indefinite integral Cos[nx]Cos[kx] about x, where both k and n are positive integer, the result is Pi when n equals to k and 0 when n is unequal to k. However, the code
sol = Integrate[Cos[n*x]*Cos[k*x], {x, -Pi, Pi},
Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result (k Sin[π k + π n] - n Sin[π k + π n] + .
k Sin[π k - π n] + n Sin[π k - π n])/(k^2 - n^2)
And then use the Simplify function,
Simplify[sol, Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result 0. Shouldn't that Integrate returns a Piecewise function like Piecewise[{{Pi, n == k}, {0, n != k}}] instead?
calculus-and-analysis
edited 2 hours ago
Mr.Wizard♦
231k294751042
asked 4 hours ago
shelure21shelure21
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
When integrate the indefinite integral Cos[nx]Cos[kx] about x, where both k and n are positive integer, the result is Pi when n equals to k and 0 when n is unequal to k. However, the code
sol = Integrate[Cos[n*x]*Cos[k*x], {x, -Pi, Pi},
Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result (k Sin[π k + π n] - n Sin[π k + π n] + .
k Sin[π k - π n] + n Sin[π k - π n])/(k^2 - n^2)
And then use the Simplify function,
Simplify[sol, Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result 0. Shouldn't that Integrate returns a Piecewise function like Piecewise[{{Pi, n == k}, {0, n != k}}] instead?
calculus-and-analysis
edited 2 hours ago
Mr.Wizard♦
231k294751042
asked 4 hours ago
shelure21shelure21
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
When integrate the indefinite integral Cos[nx]Cos[kx] about x, where both k and n are positive integer, the result is Pi when n equals to k and 0 when n is unequal to k. However, the code
sol = Integrate[Cos[n*x]*Cos[k*x], {x, -Pi, Pi},
Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result (k Sin[π k + π n] - n Sin[π k + π n] + .
k Sin[π k - π n] + n Sin[π k - π n])/(k^2 - n^2)
And then use the Simplify function,
Simplify[sol, Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result 0. Shouldn't that Integrate returns a Piecewise function like Piecewise[{{Pi, n == k}, {0, n != k}}] instead?
calculus-and-analysis
edited 2 hours ago
Mr.Wizard♦
231k294751042
asked 4 hours ago
shelure21shelure21
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
When integrate the indefinite integral Cos[nx]Cos[kx] about x, where both k and n are positive integer, the result is Pi when n equals to k and 0 when n is unequal to k. However, the code
sol = Integrate[Cos[n*x]*Cos[k*x], {x, -Pi, Pi},
Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result (k Sin[π k + π n] - n Sin[π k + π n] + .
k Sin[π k - π n] + n Sin[π k - π n])/(k^2 - n^2)
And then use the Simplify function,
Simplify[sol, Assumptions -> n ∈ Integers && k ∈ Integers && n > 0 && k > 0]
gives the result 0. Shouldn't that Integrate returns a Piecewise function like Piecewise[{{Pi, n == k}, {0, n != k}}] instead?
calculus-and-analysis
calculus-and-analysis
edited 2 hours ago
Mr.Wizard♦
231k294751042
asked 4 hours ago
shelure21shelure21
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Mr.Wizard♦
231k294751042
asked 4 hours ago
shelure21shelure21
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 hours ago
Mr.Wizard♦
231k294751042
edited 2 hours ago
Mr.Wizard♦
231k294751042
edited 2 hours ago
Mr.Wizard♦
231k294751042
231k294751042
asked 4 hours ago
shelure21shelure21
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
shelure21shelure21
184
asked 4 hours ago
shelure21shelure21
184
184
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
shelure21 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is well know issue. One way to handle it is
Simplify[ sol,
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k != n]
(* 0 *)
And
Simplify[ Limit[sol, k -> n],
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k == n ]
(* Pi *)
See
should-integrate-detect-orthogonality-of-functions-in-the-integrand
And
What assumptions to use to check for orthogonality
And
should-integrate-have-given-zero-for-this-integral
And
proper-way-to-simplify-integral-result-in-mathematica-given-integer-constraints
And
usage-of-assuming-for-integration
answered 3 hours ago
NasserNasser
57.7k488205
$endgroup$
$begingroup$
You can shorten theLimittoLimit[sol, k -> n, Assumptions -> Element[n, Integers]]
$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
add a comment |
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: "387"
};
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
shelure21 is a new contributor. Be nice, and check out our Code of Conduct.
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%2fmathematica.stackexchange.com%2fquestions%2f190139%2fa-problem-when-integrate-cosnxcoskx%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is well know issue. One way to handle it is
Simplify[ sol,
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k != n]
(* 0 *)
And
Simplify[ Limit[sol, k -> n],
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k == n ]
(* Pi *)
See
should-integrate-detect-orthogonality-of-functions-in-the-integrand
And
What assumptions to use to check for orthogonality
And
should-integrate-have-given-zero-for-this-integral
And
proper-way-to-simplify-integral-result-in-mathematica-given-integer-constraints
And
usage-of-assuming-for-integration
answered 3 hours ago
NasserNasser
57.7k488205
$endgroup$
$begingroup$
You can shorten theLimittoLimit[sol, k -> n, Assumptions -> Element[n, Integers]]
$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
add a comment |
$begingroup$
This is well know issue. One way to handle it is
Simplify[ sol,
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k != n]
(* 0 *)
And
Simplify[ Limit[sol, k -> n],
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k == n ]
(* Pi *)
See
should-integrate-detect-orthogonality-of-functions-in-the-integrand
And
What assumptions to use to check for orthogonality
And
should-integrate-have-given-zero-for-this-integral
And
proper-way-to-simplify-integral-result-in-mathematica-given-integer-constraints
And
usage-of-assuming-for-integration
answered 3 hours ago
NasserNasser
57.7k488205
$endgroup$
$begingroup$
You can shorten theLimittoLimit[sol, k -> n, Assumptions -> Element[n, Integers]]
$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
add a comment |
$begingroup$
This is well know issue. One way to handle it is
Simplify[ sol,
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k != n]
(* 0 *)
And
Simplify[ Limit[sol, k -> n],
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k == n ]
(* Pi *)
See
should-integrate-detect-orthogonality-of-functions-in-the-integrand
And
What assumptions to use to check for orthogonality
And
should-integrate-have-given-zero-for-this-integral
And
proper-way-to-simplify-integral-result-in-mathematica-given-integer-constraints
And
usage-of-assuming-for-integration
answered 3 hours ago
NasserNasser
57.7k488205
$endgroup$
This is well know issue. One way to handle it is
Simplify[ sol,
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k != n]
(* 0 *)
And
Simplify[ Limit[sol, k -> n],
Assumptions -> Element[n, Integers] && Element[k, Integers] && n > 0 && k > 0 && k == n ]
(* Pi *)
See
should-integrate-detect-orthogonality-of-functions-in-the-integrand
And
What assumptions to use to check for orthogonality
And
should-integrate-have-given-zero-for-this-integral
And
proper-way-to-simplify-integral-result-in-mathematica-given-integer-constraints
And
usage-of-assuming-for-integration
answered 3 hours ago
NasserNasser
57.7k488205
answered 3 hours ago
NasserNasser
57.7k488205
answered 3 hours ago
NasserNasser
57.7k488205
answered 3 hours ago
NasserNasser
57.7k488205
57.7k488205
$begingroup$
You can shorten theLimittoLimit[sol, k -> n, Assumptions -> Element[n, Integers]]
$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
add a comment |
$begingroup$
You can shorten theLimittoLimit[sol, k -> n, Assumptions -> Element[n, Integers]]
$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
$begingroup$
You can shorten the
Limit to Limit[sol, k -> n, Assumptions -> Element[n, Integers]]$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
You can shorten the
Limit to Limit[sol, k -> n, Assumptions -> Element[n, Integers]]$endgroup$
– Bob Hanlon
36 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
$begingroup$
@BobHanlon thanks. I am sure you are right. I was only copying what the OP had in there. But good point.
$endgroup$
– Nasser
21 mins ago
add a comment |
shelure21 is a new contributor. Be nice, and check out our Code of Conduct.
shelure21 is a new contributor. Be nice, and check out our Code of Conduct.
shelure21 is a new contributor. Be nice, and check out our Code of Conduct.
shelure21 is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Mathematica 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%2fmathematica.stackexchange.com%2fquestions%2f190139%2fa-problem-when-integrate-cosnxcoskx%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
