Are there any identities for incomplete elliptic integrals of the third kind with complex arguments?
$begingroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
333
$endgroup$
add a comment |
$begingroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
333
$endgroup$
add a comment |
$begingroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
333
$endgroup$
Abramowitz and Stegun provide identites for dealing with incomplete elliptic integrals of the first and second kinds with complex arguments:
For $tantheta = sinhphi$
17.4.8
$F(iphi | alpha) = iF(theta | frac{pi}{2} - alpha)$
17.4.9
$E(iphi | alpha) = -i E(theta | frac{pi}{2} - alpha) + i F(theta frac{pi}{2} - alpha) + itantheta(1-cos^2alpha sin^2theta)^{1/2}$
Are there any useful identities for incomplete elliptic integrals of the third kind, $Pi(n;phi | alpha)$?
I would like to evaluate such integrals numerically, but many libraries such as GSL only handle real arguments.
complex-numbers elliptic-integrals
complex-numbers elliptic-integrals
asked Jan 11 at 19:22
DannyDanny
333
asked Jan 11 at 19:22
DannyDanny
333
asked Jan 11 at 19:22
DannyDanny
333
asked Jan 11 at 19:22
DannyDanny
333
asked Jan 11 at 19:22
DannyDanny
333
333
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%2f3070249%2fare-there-any-identities-for-incomplete-elliptic-integrals-of-the-third-kind-wit%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%2f3070249%2fare-there-any-identities-for-incomplete-elliptic-integrals-of-the-third-kind-wit%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
