Update README.md

README.md

@@ -10,3 +10,27 @@ JacobiElliptic is an implementation of Toshio Fukushima's algorithms for calcula
 ## Repo Status
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+
+## Incomplete Elliptic Integrals
+|Function | Definition |
+| --- | --- |
+| `F(φ, m)` | $F(\phi\|m)$: Incomplete elliptic integral of the first kind|
+| `E(φ, m)` |  $E(\phi\|m)$: Incomplete elliptic integral of the second kind |
+| `Pi(n, φ, m)` | $\Pi(n;\\,\phi\|\\, m)$: Incomplete elliptic integral of the third kind|
+| `J(n, φ, m)` | $J (n;\\, \phi \|\\,m)=\frac{\Pi(n;\\,\phi\|\\, m) - F(\phi,m)}{n}$: Associated incomplete elliptic integral of the third kind|
+
+## Complete Elliptic Integrals
+|Function | Definition |
+| --- | --- |
+| `K(m)` | $K(m)$: Complete elliptic integral of the first kind|
+| `E(m)` |  $E(m)$: Complete elliptic integral of the second kind |
+| `Pi(n, m)` | $\Pi(n\|\\, m)$: Complete elliptic integral of the third kind|
+| `J(n, m)` | $J (n\|\\,m)=\frac{\Pi(n\|\\, m) - K(m)}{n}$: Associated incomplete elliptic integral of the third kind|
+
+## Jacobi Elliptic Functions
+|Function | Definition |
+| --- | --- |
+| `sn(u, m)` | $\text{sn}(u \| m) = \sin(\text{am}(u \| m))$ |
+| `cn(u, m)` | $\text{cn}(u \| m) = \cos(\text{am}(u \| m))$ |
+| `asn(u, m)` | $\text{asn}(u \| m) = \text{sn}^{-1}(u \| m)$ |
+| `acn(u, m)` | $\text{cn}(u \| m) = \text{cn}^{-1}(\text{am}(u \| m))$ |