Lemniscate elliptic functions: Difference between revisions

Revision as of 01:07, 26 December 2024

Revision as of 01:13, 26 December 2024

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

<math display="block">

{| class="wikitable"

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

!<math>z</math>

z

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

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

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

!<math>& \operatorname{tlhctlh} z = \sin_cos_{4} z</math>

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

|-

\\

|<math> 0</math>

\hline

|<math> \infty</math>

0

|<math> 0</math>

& \infty

|<math> 1</math>

& 0

|<math> 0</math>

& 1

|-

& 0

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

\\

|<math> 1</math>

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

|<math> 1</math>

& 1

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

& 1

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

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

|-

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

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

\\

|<math> 0</math>

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

|<math> \infty</math>

& 0

|<math> 0</math>

& \infty

|<math> 1</math>

& 0

|-

& 1

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

\\

|<math> -1</math>

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

|<math> -1</math>

& -1

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

& -1

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

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

|-

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

|<math> \sigma</math>

\\

|<math> \infty</math>

\sigma

|<math> 0</math>

& \infty

|<math> -1</math>

& 0

|<math> 0</math>

& -1

|}

& 0

\end{array}

!<math>z</math>

=== Combination and halving theorems ===