diff --git a/CHANGELOG.md b/CHANGELOG.md
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,3 +1,10 @@
+## 0.2.1.0 - 2023-10-18
+
+- Jacobi theta function with characteristics.
+
+- Derivative at 0 of the first Jacobi theta function.
+
+
 ## 0.2.0.0 - 2023-10-16
 
 New, better implementation of the Jacobi theta functions.
diff --git a/jacobi-theta.cabal b/jacobi-theta.cabal
--- a/jacobi-theta.cabal
+++ b/jacobi-theta.cabal
@@ -1,5 +1,5 @@
 name:                jacobi-theta
-version:             0.2.0.0
+version:             0.2.1.0
 synopsis:            Jacobi Theta Functions
 description:         Evaluation of the Jacobi theta functions.
 homepage:            https://github.com/stla/jacobi-theta#readme
diff --git a/src/Math/JacobiTheta.hs b/src/Math/JacobiTheta.hs
--- a/src/Math/JacobiTheta.hs
+++ b/src/Math/JacobiTheta.hs
@@ -1,9 +1,23 @@
+{-|
+Module      : Math.JacobiTheta
+Description : Jacobi theta functions.
+Copyright   : (c) Stéphane Laurent, 2023
+License     : BSD3
+Maintainer  : laurent_step@outlook.fr
+
+Provides the four usual Jacobi theta functions, the Jacobi theta function 
+with characteristics, the derivative of the first Jacobi theta function, 
+as well as a function for the derivative at @0@ only of the first Jacobi 
+theta function.
+-}
 module Math.JacobiTheta
   (
     jtheta1,
     jtheta2,
     jtheta3,
     jtheta4,
+    jthetaAB,
+    jtheta1Dash0,
     jtheta1Dash 
   )
   where
@@ -112,7 +126,7 @@
 ljtheta1 :: Cplx -> Cplx -> Cplx
 ljtheta1 z tau = ljtheta2 (z - 0.5) tau
 
--- | First Jacobi theta function
+-- | First Jacobi theta function.
 jtheta1 ::
      Complex Double -- ^ z
   -> Complex Double -- ^ q, the nome
@@ -125,7 +139,7 @@
 ljtheta2 z tau = 
   funM z tau + dologtheta3 (z + 0.5 * tau) tau 0 1000
 
--- | Second Jacobi theta function
+-- | Second Jacobi theta function.
 jtheta2 ::
      Complex Double -- ^ z
   -> Complex Double -- ^ q, the nome
@@ -134,7 +148,7 @@
   where
     tau = getTauFromQ q
 
--- | Third Jacobi theta function
+-- | Third Jacobi theta function.
 jtheta3 ::
      Complex Double -- ^ z
   -> Complex Double -- ^ q, the nome
@@ -143,7 +157,7 @@
   where
     tau = getTauFromQ q
 
--- | Fourth Jacobi theta function
+-- | Fourth Jacobi theta function.
 jtheta4 ::
      Complex Double -- ^ z
   -> Complex Double -- ^ q, the nome
@@ -152,7 +166,55 @@
   where
     tau = getTauFromQ q
 
--- | Derivative of the first Jacobi theta function
+-- | Jacobi theta function with characteristics. This is a family of functions, 
+--  containing the first Jacobi theta function (@a=b=0.5@), the second Jacobi 
+--  theta function (@a=0.5, b=0@), the third Jacobi theta function (@a=b=0@)
+--  and the fourth Jacobi theta function (@a=0, b=0.5@). The examples given 
+--  below show the periodicity-like properties of these functions:
+--  
+-- >>> import Data.Complex
+-- >>> a = 2 :+ 0.3
+-- >>> b = 1 :+ (-0.6)
+-- >>> z = 0.1 :+ 0.4
+-- >>> tau = 0.2 :+ 0.3
+-- >>> im = 0 :+ 1 
+-- >>> q = exp(im * pi * tau)
+-- >>> jab = jthetaAB a b z q
+-- >>> jthetaAB a b (z + pi) q
+-- (-5.285746223832433e-3) :+ 0.1674462628348814
+-- 
+-- >>> jab * exp(2 * im * pi * a)
+-- (-5.285746223831987e-3) :+ 0.16744626283488154
+-- 
+-- >>> jtheta_ab a b (z + pi*tau) q
+-- 0.10389127606987271 :+ 0.10155646232306936
+-- 
+-- >>> jab * exp(-im * (pi*tau + 2*z + 2*pi*b))
+-- 0.10389127606987278 :+ 0.10155646232306961
+jthetaAB ::
+     Complex Double -- ^ characteristic a
+  -> Complex Double -- ^ characteristic b
+  -> Complex Double -- ^ z
+  -> Complex Double -- ^ q, the nome
+  -> Complex Double
+jthetaAB a b z q = c * jtheta3 (alpha + beta) q
+  where
+    tau = getTauFromQ q
+    alpha = pi * a * tau
+    beta  = z + pi * b
+    c     =  exp(i_ * a * (alpha + 2*beta)) 
+
+-- | Derivative at 0 of the first Jacobi theta function. This is much more 
+--  efficient than evaluating @jtheta1Dash@ at @0@.
+jtheta1Dash0 :: 
+     Complex Double -- ^ q, the nome
+  -> Complex Double
+jtheta1Dash0 q = 
+  -2 * i_ * jab * jab * jab
+  where
+    jab = jthetaAB (1/6) 0.5 0 (q*q*q)
+
+-- | Derivative of the first Jacobi theta function.
 jtheta1Dash :: 
      Complex Double -- ^ z
   -> Complex Double -- ^ q, the nome
diff --git a/tests/Main.hs b/tests/Main.hs
--- a/tests/Main.hs
+++ b/tests/Main.hs
@@ -2,7 +2,7 @@
 import Approx ( approx )
 import Data.Complex ( Complex(..) )
 import Math.JacobiTheta
-    ( jtheta1, jtheta2, jtheta3, jtheta4, jtheta1Dash )
+    ( jtheta1, jtheta2, jtheta3, jtheta4, jtheta1Dash, jtheta1Dash0, jthetaAB )
 import           Test.Tasty       (defaultMain, testGroup)
 import           Test.Tasty.HUnit (assertEqual, testCase)
 
@@ -90,6 +90,23 @@
           expected = 27.746815969548447 :+ 31.241216782108797
       assertEqual ""
         (approx 10 obtained)
-        (approx 10 expected)      
+        (approx 10 expected),
+
+    testCase "jtheta1Dash at 0" $ do
+      let q''' = 0.556 :+ 0.283
+          expected = jtheta1Dash0 q'''
+          obtained = jtheta1Dash 0 q'''
+      assertEqual ""
+        (approx 10 obtained)
+        (approx 10 expected),
+
+    testCase "A formula involving some jthetaAB at z=0" $ do
+      let q''' = 0.556 :+ 0.283
+          expected = (jthetaAB (1/6) 0.5 0 q''')**3
+          obtained = 
+            (jthetaAB (1/6) (1/6) 0 q''')**3 + (jthetaAB (1/6) (5/6) 0 q''')**3
+      assertEqual ""
+        (approx 10 obtained)
+        (approx 10 expected)
 
   ]