Lemniscate elliptic functions: Difference between revisions

Revision as of 15:58, 7 December 2024

Revision as of 01:07, 26 December 2024

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which can be compared to the cyclotomic analog

:<math>\operatorname{deg}\Phi_{k}=k\prod_{p|k}\left(1-\frac{1}{p}\right).</math>

===Specific values===

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Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into {{math|''n''}} parts of equal length, using only basic arithmetic and square roots, if and only if {{math|''n''}} is of the form <math>n = 2^kp_1p_2\cdots p_m</math> where {{math|''k''}} is a non-negative [[integer]] and each {{math|''p''_''i''}} (if any) is a distinct [[Fermat prime]].<ref>{{harvp|Rosen|1981}}</ref>

<math display="block">

{| class="wikitable"

\begin{array}{|c|cc|}

|-

! <math>n</math> !!& <math>\operatorname{cl}n\varpi</math> !!& <math>\operatorname{sl}n\varpi</math>

\\

|-

\hline

| <math> 1</math>

1

| <math> -1</math>

& -1

| <math> 0</math>

& 0

|-

\\

|<math> \tfrac{5}{6}</math>

|<math> -\sqrt[4]{2\sqrt{3}-3}</math>

|<math>& \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}2\bigr)</math>sqrt{3}-3}

|& <math> \tfrac12\biglleft(\sqrt{3}+1-\sqrt[4]{12}\bigrright)</math>

|-

\\

|<math> \tfrac{3}{4}</math>

|<math> -\sqrt{\sqrt2-1}</math>

|<math>& -\sqrt{\sqrt2-1}</math>

|<math>& -\sqrt{\sqrt2-1}</math>

|-

\\

| <math> \tfrac{2}{3}</math>

| <math> -\tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>

|& <math> -\sqrt[4]{2tfrac12\left(\sqrt{3}+1-3\sqrt[4]{12}</math>\right)

|<math>& -\sqrt[4]{2\sqrt{3}-3}</math>

|-

\\

| <math> \tfrac{1}{2}</math>

| <math> 0</math>

& 0

| <math> 1</math>

& 1

|-

\\

| <math> \tfrac{1}{3}</math>

| <math> \tfrac12\bigl(\sqrt{3}+1-\sqrt[4]{12}\bigr)</math>

|& <math> \sqrt[4]{2tfrac12\left(\sqrt{3}+1-3\sqrt[4]{12}</math>\right)

|<math>& \sqrt[4]{2\sqrt{3}-3}</math>

|-

\\

| <math> \tfrac{1}{4}</math>

| <math> \sqrt{\sqrt2-1}</math>

|& <math> \sqrt{\sqrt2-1}</math>

|& <math> \sqrt{\sqrt2-1}</math>

|-

\\

|<math> \tfrac{1}{6}</math>

|<math> \sqrt[4]{2\sqrt{3}-3}</math>

|<math>& \tfrac12\bigl(sqrt[4]{2\sqrt{3}+1-\sqrt[4]{123}\bigr)</math>

|& <math> -\tfrac12\biglleft(\sqrt{3}+1-\sqrt[4]{12}\bigrright)</math>

|}

\end{array}

| <math> 1</math>

== Relation to geometric shapes ==