# Schwarzian derivative

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##### 1: 1.13 Differential Equations
βΊA solution becomes unique, for example, when $w$ and $\ifrac{\mathrm{d}w}{\mathrm{d}z}$ are prescribed at a point in $D$. … βΊ
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###### Elimination of First Derivative by Change of Independent Variable
βΊHere dots denote differentiations with respect to $\zeta$, and $\left\{z,\zeta\right\}$ is the Schwarzian derivative: … βΊ
##### 2: 4.20 Derivatives and Differential Equations
###### §4.20 Derivatives and Differential Equations
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4.20.9 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w=0,$
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4.20.10 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a^{2}w^{2}=1,$
##### 3: 4.34 Derivatives and Differential Equations
###### §4.34 Derivatives and Differential Equations
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4.34.7 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-a^{2}w=0,$
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4.34.8 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=1,$
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4.34.9 $\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}-a^{2}w^{2}=-1,$
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4.34.10 $\frac{\mathrm{d}w}{\mathrm{d}z}+a^{2}w^{2}=1,$
##### 4: 4.7 Derivatives and Differential Equations
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###### §4.7(i) Logarithms
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4.7.5 $\frac{\mathrm{d}w}{\mathrm{d}z}=\frac{f^{\prime}(z)}{f(z)}$
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##### 5: 22.13 Derivatives and Differential Equations
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###### §22.13(i) Derivatives
βΊ βΊNote that each derivative in Table 22.13.1 is a constant multiple of the product of the corresponding copolar functions. … βΊ
22.13.1 $\left(\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{sn}\left(z,k\right)\right)^{% 2}=\left(1-{\operatorname{sn}}^{2}\left(z,k\right)\right)\left(1-k^{2}{% \operatorname{sn}}^{2}\left(z,k\right)\right),$
##### 6: 7.10 Derivatives
###### §7.10 Derivatives
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7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.
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$\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}=-\pi z\mathrm{g}\left(z% \right),$
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$\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}=\pi z\mathrm{f}\left(z% \right)-1.$
##### 7: 1.5 Calculus of Two or More Variables
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###### §1.5(i) Partial Derivatives
βΊThe function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, $\,{\partial}^{2}f/\,\partial x\,\partial y$, and $\,{\partial}^{2}f/\,\partial y\,\partial x$ are continuous. … βΊ
###### Chain Rule
βΊSuppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times[c,d)$. … βΊ
##### 8: 36.10 Differential Equations
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###### §36.10(ii) Partial Derivatives with Respect to the $x_{n}$
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36.10.7 $\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$
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36.10.8 $\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{3}}{{\partial y}^{n}},$
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36.10.10 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}=i^{n}\frac{{\partial}^{2n}% \Psi_{3}}{{\partial z}^{2n}}.$
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##### 9: 19.18 Derivatives and Differential Equations
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###### §19.18(i) Derivatives
βΊLet $\partial_{j}=\ifrac{\partial}{\partial z_{j}}$, and $\mathbf{e}_{j}$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … βΊ
19.18.14 $\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial w}{\partial y}.$
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19.18.15 $\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{% 2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}.$
##### 10: 30.12 Generalized and Coulomb Spheroidal Functions
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30.12.1 $\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac{\mathrm{d}w}{\mathrm{d}z}% \right)+{\left(\lambda+\alpha z+\gamma^{2}(1-z^{2})-\frac{\mu^{2}}{1-z^{2}}% \right)w}=0,$
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30.12.2 $\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac{\mathrm{d}w}{\mathrm{d}z}% \right)+\left(\lambda+\gamma^{2}(1-z^{2})-\frac{\alpha(\alpha+1)}{z^{2}}-\frac% {\mu^{2}}{1-z^{2}}\right)w=0,$