One entry per mathematical family, deduplicated across naming variants and grouped by type. Parametric families collapse many historical singletons into a single entity with a parameter space.
| Koebe z/(1-z)² | z/(1-z)^2 | 1 · 8 | |
| Koebe Kα: z/(1-z)^(2α) | z/(1-z)^(2*alpha) | parametric (1) | 16 · 25 |
| z(1-z/2)/(1-z)^2 | z*(1 - z/2)/(1-z)**2 | 1 · 1 | |
| z(1-z/3)/(1-z)^2 | z*(1 - z/3)/(1-z)**2 | 1 · 1 | |
| z(1+z/2)/(1-z) | z*(1 + z/2)/(1 - z) | 1 · 1 | |
| z(1+z)/(1-z) | z*(1 + z)/(1 - z) | 1 · 1 | |
| (z - z^2/2)/(1-z)^2 | (z - z**2/2)/(1-z)**2 | 1 · 1 | |
| z/(1-z) | z/(1-z) | 1 · 8 | |
| z/(1-z/2) | z/(1 - z/2) | 1 · 8 | |
| z/(1-z/2)² | z/(1 - z/2)^2 | 1 · 8 | |
| z/(1-z/3) | z/(1 - z/3) | 1 · 8 | |
| z/(1-z/4) | z/(1 - z/4) | 1 · 1 | |
| z/(1-z)^4 | z/(1-z)**4 | 1 · 1 | |
| z/(1-z/5) | z/(1 - z/5) | 1 · 1 | |
| z/(1-z)^3 | z/(1-z)**3 | 1 · 1 | |
| z/(1-z+z^2/2) | z/(1 - z + z**2/2) | 1 · 1 | |
| z/(1-z+z^2) | z/(1-z+z**2) | 1 · 1 | |
| z/(1-z^2) | z/(1-z**2) | 1 · 1 | |
| z/(1+z/2) | z/(1 + z/2) | 1 · 1 | |
| z/(1+z/3) | z/(1 + z/3) | 1 · 1 | |
| z/(1+z) | z/(1+z) | 1 · 8 | |
| z/(1+z²) | z/(1+z^2) | 1 · 8 | |
| z/(1+z)^3 | z/(1+z)**3 | 1 · 1 | |
| z/(1+z+z^2) | z/(1+z+z**2) | 1 · 1 | |
| z/(1+z)^2 | z/(1+z)**2 | 1 · 1 |
| z + 3z² (truly disproven) | z + 3*z**2 | 1 · 5 | |
| z − z²/4 | - | 1 · 4 | |
| z + z²/2 | - | 1 · 4 | |
| z + z²/4 | - | 1 · 4 | |
| z + z²/4 + z³/9 | - | 1 · 4 | |
| z + z² + z³ | - | 1 · 4 | |
| z + z³/3 | - | 1 · 4 | |
| z - z^2/2 | z - z**2/2 | 1 · 1 | |
| z - z^2/2 + z^3/3 | z - z**2/2 + z**3/3 | 1 · 1 | |
| z - z^2/4 | z - z**2/4 | 1 · 1 | |
| z - z^3/3 | z - z**3/3 | 1 · 1 | |
| z + z^2/5 | z + z**2/5 | 1 · 1 | |
| z + z^5/5 | z + z**5/5 | 1 · 1 | |
| z + z^2/2 | z + z**2/2 | 1 · 1 | |
| z + z^2/2 + z^3/3 | z + z**2/2 + z**3/3 | 1 · 1 | |
| z + z^2/4 + z^3/9 | z + z**2/4 + z**3/9 | 1 · 1 | |
| z + z^3/4 | z + z**3/4 | 1 · 1 | |
| z + z^4/4 | z + z**4/4 | 1 · 1 | |
| z + z^2/6 | z + z**2/6 | 1 · 1 | |
| z + z^2/3 | z + z**2/3 | 1 · 1 | |
| z + z^3/3 | z + z**3/3 | 1 · 1 | |
| z + z²/4 (C01 boundary) | z + z**2/4 | 1 · 5 |
| (e^(2z)-1)/2 | (exp(2*z) - 1)/2 | 1 · 1 | |
| (e^(3z)-1)/3 | (exp(3*z) - 1)/3 | 1 · 1 | |
| z cosh(z) | z*cosh(z) | 1 · 8 | |
| z·exp(2z) — convex of order 1/2 | z*exp(2*z) | 1 · 5 | |
| z e^(-z^2/2) | z*exp(-z**2/2) | 1 · 1 | |
| z e^(-z^2/3) | z*exp(-z**2/3) | 1 · 1 | |
| z e^(-z/2) | z*exp(-z/2) | 1 · 1 | |
| z e^(-z/3) | z*exp(-z/3) | 1 · 1 | |
| z e^(-z) | z*exp(-z) | 1 · 8 | |
| z e^z | z*exp(z) | 1 · 8 | |
| z e^(z²) | z*exp(z/2) | 1 · 8 | |
| z e^(z^2/2) | z*exp(z**2/2) | 1 · 1 | |
| z e^(z^2/4) | z*exp(z**2/4) | 1 · 1 | |
| z e^(z^2/6) | z*exp(z**2/6) | 1 · 1 | |
| z·exp(z³/3) | z*exp(z**3/3) | 1 · 5 | |
| z·exp(z/2) — half exponential | z*exp(z/2) | 1 · 5 | |
| z e^(z/3) | z*exp(z/3) | 1 · 1 | |
| z e^(z/4) | z*exp(z/4) | 1 · 1 | |
| sinh(z) (normalized) | sinh(z) | 1 · 8 |
| arctanh(z) | atanh(z) | 1 · 8 | |
| sin(z) | sin(z) | 1 · 8 | |
| tan(z) | tan(z) | 1 · 1 | |
| 3 sin(z/3) | 3*sin(z/3) | 1 · 1 | |
| 2 sin(z/2) | 2*sin(z/2) | 1 · 1 | |
| 2 tan(z/2) | 2*tan(z/2) | 1 · 1 | |
| arctan(z) (normalized) | atan(z) | 1 · 5 | |
| z cos(z) | z*cos(z) | 1 · 8 | |
| z cos(z/2) | z*cos(z/2) | 1 · 1 | |
| z cos(z/3) | z*cos(z/3) | 1 · 1 | |
| z sec(z) | z/cos(z) | 1 · 1 | |
| z sec(z/2) | z/cos(z/2) | 1 · 1 |
| tanh(z) | tanh(z) | 1 · 1 | |
| 3 sinh(z/3) | 3*sinh(z/3) | 1 · 1 | |
| 3 tanh(z/3) | 3*tanh(z/3) | 1 · 1 | |
| 2 tanh(z/2) | 2*tanh(z/2) | 1 · 1 | |
| z cosh(z/2) | z*cosh(z/2) | 1 · 1 | |
| z cosh(z/3) | z*cosh(z/3) | 1 · 1 | |
| z sech(z) | z/cosh(z) | 1 · 1 |
| z + 2z² | z + 2z^2 | 1 · 4 | |
| z + z² | z + z^2 | 1 · 8 | |
| z + z² + z³/2 | z + z^2 + z^3/2 | 1 · 8 |
| Identity z | z | 1 · 8 | |
| −log(1−z) | -log(1-z) | 1 · 8 |
| log(1+z) | log(1 + z) | 1 · 1 | |
| -2 log(1-z/2) | -2*log(1 - z/2) | 1 · 1 |
| Bi-univalent √2 extremal | - | 1 · 4 |
| Convex of order α — K(α) | - | parametric (1) | 25 · 40 |
| erf(z) normalized | - | 1 · 4 |
| Laguerre L_n | - | parametric (1) | 10 · 19 |
| Two-parameter Mittag-Leffler E_{a,b} | - | parametric (2) | 36 · 36 |
| Strongly starlike SS*(β) | - | parametric (1) | 20 · 35 |
| q-exponential e_q(z) | - | parametric (1) | 38 · 71 |
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