diff --git a/CHANGELOG.md b/CHANGELOG.md
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,5 +1,11 @@
 # hypergeometric
 
+## 0.1.5.0
+
+  * Throw errors when outside of the radius of convergence for a series
+  * Add `euler` which transforms ₂F₁ to something that converges for all z.
+  * Fix egregious typo in documentation
+
 ## 0.1.4.0
 
   * Add `agm`, `completeElliptic`
diff --git a/hypergeometric.cabal b/hypergeometric.cabal
--- a/hypergeometric.cabal
+++ b/hypergeometric.cabal
@@ -1,6 +1,6 @@
 cabal-version:   1.18
 name:            hypergeometric
-version:         0.1.4.0
+version:         0.1.5.0
 license:         AGPL-3
 license-file:    COPYING
 copyright:       Copyright: (c) 2022 Vanessa McHale
@@ -9,7 +9,7 @@
 bug-reports:     https://github.com/vmchale/hypergeometric/issues
 synopsis:        Hypergeometric functions
 description:
-    Haskell implementation of hypergeometric functions and associated statistical and special functions, viz. erf, normal cdf, incomplete beta, F-distribution cdf, \(\chi^2\)-distribution cdf, t-distrubtion cdf.
+    Haskell implementation of hypergeometric functions and associated statistical and special functions: erf, normal cdf, incomplete beta, regularized beta, F-distribution cdf, \(\chi^2\)-distribution cdf, t-distrubtion cdf.
     Also includes Lanczos' approximation of the gamma function.
 
 category:        Math, Statistics
diff --git a/src/Math/Hypergeometric.hs b/src/Math/Hypergeometric.hs
--- a/src/Math/Hypergeometric.hs
+++ b/src/Math/Hypergeometric.hs
@@ -1,5 +1,6 @@
--- | See McHale, Vanessa [\"Hypergeometric Functions for Statistical Computing\"](http://vmchale.com/static/serve/hypergeometric.pdf) and especially Shaw, Ernest [\"Hypergeometric Functions and CDFs in J\"](https://www.jsoftware.com/papers/jhyper.pdf)
+-- | See Shaw, Ewart [\"Hypergeometric Functions and CDFs in J\"](https://www.jsoftware.com/papers/jhyper.pdf).
 module Math.Hypergeometric ( hypergeometric
+                           , euler
                            , erf
                            , ncdf
                            ) where
@@ -16,27 +17,56 @@
 {-# SPECIALIZE ncdf :: Double -> Double #-}
 {-# SPECIALIZE ncdf :: Float -> Float #-}
 -- | CDF of the standard normal \( N(0,1) \)
-ncdf :: (Eq a, Floating a) => a -> a
+ncdf :: (Ord a, Floating a) => a -> a
 ncdf z = (1/2) * (1 + erf (z / sqrt 2))
 
 {-# SPECIALIZE erf :: Double -> Double #-}
 {-# SPECIALIZE erf :: Float -> Float #-}
 -- | [erf](https://mathworld.wolfram.com/Erf.html)
-erf :: (Eq a, Floating a) => a -> a
-erf z = (2 * z * exp (-z^(2::Int)) / sqrt pi) * hypergeometric [1] [3/2] (z^(2::Int))
+erf :: (Ord a, Floating a) => a -> a
+erf z = (2 * z * exp (-(z^(2::Int))) / sqrt pi) * hypergeometric [1] [3/2] (z^(2::Int))
 
+{-# SPECIALIZE euler :: Double -> Double -> Double -> Double -> Double #-}
+{-# SPECIALIZE euler :: Float -> Float -> Float -> Float -> Float #-}
+-- | Euler's transform:
+--
+-- \( \displaystyle _2F_1(a,b;c;z) = (1-z)^{-a} {}_2F_1\left(a,c-b;c;\frac{z}{z-1}\right) \)
+--
+-- The right-hand side converges for all \(z\).
+--
+-- Koekoek, Roelef and Swarttouw, René F. [The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue](https://arxiv.org/abs/math/9602214).
+--
+-- @since 0.1.5.0
+euler :: (Ord a, Floating a)
+      => a -- ^ \(a\)
+      -> a -- ^ \(b\)
+      -> a -- ^ \(c\)
+      -> a -- ^ \(z\)
+      -> a
+euler a b c z = hypergeometric [a, c-b] [c] (z/z-1)
+
 {-# SPECIALIZE hypergeometric :: [Double] -> [Double] -> Double -> Double #-}
 {-# SPECIALIZE hypergeometric :: [Float] -> [Float] -> Float -> Float #-}
 -- | \( _pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \displaystyle\sum_{n=0}^\infty\frac{(a_1)_n\cdots(a_p)_n}{(b_1)_b\cdots(b_q)_n}\frac{z^n}{n!} \)
 --
+-- The radius of convergence is
+--
+-- \( \rho = \begin{cases} \infty & \text{if} & p<q+1 \\ 1 & \text{if} & p=q+1 \\ 0 & \text{if} & p>q+1 \\ \end{cases} \)
+--
 -- This iterates until the result stabilizes.
-hypergeometric :: (Eq a, Fractional a)
+hypergeometric :: (Ord a, Fractional a)
                => [a] -- ^ \( a_1,\ldots,a_p \)
                -> [a] -- ^ \( b_1,\ldots,b_q \)
                -> a -- ^ \( z \)
                -> a
 hypergeometric as bs z = sumUntilEq
-    [ (product (fmap (`risingFactorial` n) as) / product (fmap (`risingFactorial` n) bs)) * (z ^ n) / factorial n | n <- [0..] ]
+    [ (product (fmap (`risingFactorial` n) as) / product (fmap (`risingFactorial` n) bs)) * (zϵ ^ n) / factorial n | n <- [0..] ]
+  where
+    zϵ = 𝜌 z
+    𝜌 = case compare (length as) (length bs+1) of
+        LT -> id
+        EQ -> if z > 1 then error "Outside the radius of convergence." else id
+        GT -> if z /= 0 then error "Outside the radius of convergence." else id
 
 sumUntilEq :: (Eq a, Num a) => [a] -> a
 sumUntilEq = sumUntilEqLoop 0
diff --git a/src/Math/SpecialFunction.hs b/src/Math/SpecialFunction.hs
--- a/src/Math/SpecialFunction.hs
+++ b/src/Math/SpecialFunction.hs
@@ -1,5 +1,6 @@
 module Math.SpecialFunction ( incbeta
                             , beta
+                            , regbeta
                             , bessel1
                             , gamma
                             , gammaln
@@ -14,14 +15,14 @@
 
 {-# SPECIALIZE tcdf :: Double -> Double -> Double #-}
 {-# SPECIALIZE tcdf :: Float -> Float -> Float #-}
--- | Converges if and only if \(|x| < \sqrt{\nu} \)
+-- | Converges if and only if \(|x| \leq \sqrt{\nu} \)
 --
 -- @since 0.1.2.0
 tcdf :: (Floating a, Ord a)
      => a -- ^ \(\nu\) (degrees of freedom)
      -> a -- ^ \(x\)
      -> a
-tcdf 𝜈 x = 0.5 + x * gamma (0.5*(𝜈+1)) / (sqrt(pi*𝜈) * gamma(𝜈/2)) * hypergeometric [0.5, 0.5*(𝜈+1)] [1.5] (-x^(2::Int)/𝜈)
+tcdf 𝜈 x = 0.5 + x * gamma (0.5*(𝜈+1)) / (sqrt(pi*𝜈) * gamma(𝜈/2)) * hypergeometric [0.5, 0.5*(𝜈+1)] [1.5] (-(x^(2::Int))/𝜈)
 
 {-# SPECIALIZE bessel1 :: Double -> Double -> Double #-}
 {-# SPECIALIZE bessel1 :: Float -> Float -> Float #-}
@@ -69,15 +70,13 @@
 -- | \(a^{-1}x^a{}_1F_1(a;1+a;-x) \)
 incgamma :: (Floating a, Ord a) => a -> a -> a
 incgamma a x = (1/a) * x ** a * hypergeometric [a] [1+a] (-x)
--- TODO: writeup?
---
 -- chisqcdf 10 28 works better this way than w/ e^-x ... x
 
 -- (1 2 H. _1.1) 1 hangs indefinitely
 
 {-# SPECIALIZE incbeta :: Double -> Double -> Double -> Double #-}
 {-# SPECIALIZE incbeta :: Float -> Float -> Float -> Float #-}
--- | Incomplete beta function, \(|z|<1\)
+-- | Incomplete beta function, \(|z|\leq 1\)
 --
 -- Calculated with \(B(z;a,b)=\displaystyle\frac{z^a}{a}{}_2F_1(a, 1-b; a+1; z)\)
 --
@@ -91,8 +90,14 @@
 
 {-# SPECIALIZE regbeta :: Double -> Double -> Double -> Double #-}
 {-# SPECIALIZE regbeta :: Float -> Float -> Float -> Float #-}
--- | \(I(z;a,b) = \displaystyle\frac{B(z;a,b)}{B(a,b)}\)
-regbeta :: (Floating a, Ord a) => a -> a -> a -> a
+-- | [Regularized beta function](https://mathworld.wolfram.com/RegularizedBetaFunction.html), \(|z|\leq 1\)
+--
+-- \(I(z;a,b) = \displaystyle\frac{B(z;a,b)}{B(a,b)}\)
+regbeta :: (Floating a, Ord a) 
+        => a -- ^ \(z\)
+        -> a -- ^ \(a\)
+        -> a -- ^ \(b\)
+        -> a
 regbeta z a b = incbeta z a b / beta a b
 
 {-# SPECIALIZE fcdf :: Double -> Double -> Double -> Double #-}