Lemniscate elliptic functions: Difference between revisions

Revision as of 01:13, 26 December 2024

Revision as of 05:45, 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===

Line 298 ⟶ 296:

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>

{| class="wikitable"

<math display="block">

|-

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

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

|-

\\

| <math> 1</math>

\hline

| <math> -1</math>

1

| <math> 0</math>

& -1

|-

& 0

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

\\

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

\tfrac{5}{6}

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

& -\sqrt[4]{2\sqrt{3}-3}

|-

& \tfrac12\left(\sqrt{3}+1-\sqrt[4]{12}\right)

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

\\

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

\tfrac{3}{4}

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

|-

& \sqrt{\sqrt2-1}

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

\\

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

\tfrac{2}{3}

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

|-

& \sqrt[4]{2\sqrt{3}-3}

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

\\

| <math> 0</math>

\tfrac{1}{2}

| <math> 1</math>

& 0

|-

& 1

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

\\

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

\tfrac{1}{3}

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

|-

& \sqrt[4]{2\sqrt{3}-3}

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

\\

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

\tfrac{1}{4}

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

|-

& \sqrt{\sqrt2-1}

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

\\

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

\tfrac{1}{6}

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

|}

& \tfrac12\left(\sqrt{3}+1-\sqrt[4]{12}\right)

\end{array}

</math>

== Relation to geometric shapes ==

Line 1,011 ⟶ 1,007:

That table shows the most important values of the '''Hyperbolic Lemniscate Tangent and Cotangent''' functions:

{| class="wikitable"

<math display="block">

!<math>z</math>

\begin{array}{|c|rr|ll|}

!<math> \operatorname{clh} z</math>

z

& !<math> \operatorname{clhslh} z</math>

& !<math> \operatorname{slhctlh} z = \cos_{4} z</math>

& !<math> \operatorname{ctlhtlh} z = \cos_sin_{4} z</math>

|-

& \operatorname{tlh} z = \sin_{4} z

|<math> 0</math>

\\

|<math> \infty</math>

\hline

|<math> 0</math>

0

|<math> 1</math>

& \infty

|<math> 0</math>

& 0

|-

& 1

|<math> {\tfrac14}\sigma</math>

& 0

|<math> 1</math>

\\

|<math> 1</math>

{\tfrac14}\sigma

|<math> 1\big/\sqrt[4]{2}</math>

& 1

|<math> 1\big/\sqrt[4]{2}</math>

& 1

|-

& 1\big/\sqrt[4]{2}

|<math> {\tfrac12}\sigma</math>

& 1\big/\sqrt[4]{2}

|<math> 0</math>

\\

|<math> \infty</math>

{\tfrac12}\sigma

|<math> 0</math>

& 0

|<math> 1</math>

& \infty

|-

& 0

|<math> {\tfrac34}\sigma</math>

& 1

|<math> -1</math>

\\

|<math> -1</math>

{\tfrac34}\sigma

|<math> -1\big/\sqrt[4]{2}</math>

& -1

|<math> 1\big/\sqrt[4]{2}</math>

& -1

|-

& -1\big/\sqrt[4]{2}

|<math> \sigma</math>

& 1\big/\sqrt[4]{2}

|<math> \infty</math>

\\

|<math> 0</math>

\sigma

|<math> -1</math>

& \infty

|<math> 0</math>

& 0

|}

& -1

& 0

\end{array}

</math>

=== Combination and halving theorems ===