Merge pull request #73 from nschloe/gudermannian

add gudermannian

Author: nschloe

.pre-commit-config.yaml

@@ -1,6 +1,6 @@
 repos:
   - repo: https://github.com/PyCQA/isort
-    rev: 5.11.2
+    rev: 5.11.4
     hooks:
       - id: isort
 

README.md

@@ -188,6 +188,10 @@ All plots are created with default settings.
 | :------------------------------------------------------------------------: | :------------------------------------------------------------: | :--------------------------------------------------------------------: |
 | [Integral sine _Si_](https://en.wikipedia.org/wiki/Trigonometric_integral) |                      Integral cosine _Ci_                      | [Lambert W function](https://en.wikipedia.org/wiki/Lambert_W_function) |
 
+|   <img src="https://nschloe.github.io/cplot/gudermannian.png" width="70%">   | <img src="https://nschloe.github.io/cplot/exp1.png" width="70%"> | <img src="https://nschloe.github.io/cplot/expi.png" width="70%"> |
+| :--------------------------------------------------------------------------: | :--------------------------------------------------------------: | :--------------------------------------------------------------: |
+| [Gudermannian function](https://en.wikipedia.org/wiki/Gudermannian_function) |                     Exponential integral E1                      |                     Exponential integral Ei                      |
+
 |   <img src="https://nschloe.github.io/cplot/zeta.png" width="70%">   | <img src="https://nschloe.github.io/cplot/bernoulli.png" width="70%"> |   <img src="https://nschloe.github.io/cplot/dirichlet-eta.png" width="70%">    |
 | :------------------------------------------------------------------: | :-------------------------------------------------------------------: | :----------------------------------------------------------------------------: |
 | [`mpmath.zeta`](https://en.wikipedia.org/wiki/Riemann_zeta_function) |                          Bernoulli function                           | [Dirichlet eta function](https://en.wikipedia.org/wiki/Dirichlet_eta_function) |

tests/generate-readme-figures.py

@@ -65,6 +65,14 @@ def hurwitz_zeta(s, a):
     return np.reshape(out, s.shape)
 
 
+def gudermannian(z):
+    return 2 * np.arctan(np.tanh(0.5 * z))
+
+
+def gudermannian_inv(z):
+    return 2 * np.arctanh(np.tan(0.5 * z))
+
+
 def hurwitz_zeta_a(s, a):
     """
     Like hurwitz_zeta(), but with the vectorization in the second component.
@@ -292,10 +300,13 @@ def bernoulli(z):
     #
     ("si.png", lambda z: sici(z)[0], (-15, +15), (-15, +15)),
     ("ci.png", lambda z: sici(z)[1], (-15, +15), (-15, +15)),
-    ("expi.png", expi, (-15, +15), (-15, +15)),
+    ("lambertw.png", lambertw, (-5, +5), (-5, +5)),
+    #
     #
+    ("gudermannian.png", gudermannian, (-10, 10), (-10, 10)),
+    # ("gudermannian_inv.png", gudermannian_inv, (-2, 2), (-2, 2)),
     ("exp1.png", exp1, (-5, +5), (-5, +5)),
-    ("lambertw.png", lambertw, (-5, +5), (-5, +5)),
+    ("expi.png", expi, (-15, +15), (-15, +15)),
     #
     ("zeta.png", zeta, (-30, +30), (-30, +30)),
     ("bernoulli.png", bernoulli, (-30, +30), (-30, +30)),