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21: 8.13 Zeros

… ►For and $n=1,2,3,\mathrm{\dots }$, there exist: …

22: 31.14 General Fuchsian Equation

… ►The algorithm returns a list of solutions if they exist. …

23: 35.2 Laplace Transform

… ►Suppose there exists a constant ${𝐗}_0\in 𝛀$ such that for all $𝐗\in 𝛀$. …

24: Mathematical Introduction

… ►The exceptions are ones for which the existing notations have drawbacks. … ►Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …

25: 1.13 Differential Equations

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§1.13(i) Existence of Solutions

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26: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty }$, which are branch points (or poles) of the functions, in general. …

27: 27.8 Dirichlet Characters

… ►A Dirichlet character $\chi (modk)$ is called primitive (mod $k$) if for every proper divisor $d$ of $k$ (that is, a divisor ), there exists an integer $a\equiv 1(modd)$, with $\left(a,k\right)=1$ and $\chi \left(a\right)\ne 1$. …

28: 31.8 Solutions via Quadratures

… ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ may become a rational function in $z$. …

29: 1.9 Calculus of a Complex Variable

… ►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: … ►If $f^{\prime }(z)$ exists at $z=x+\mathrm{i}y$ and $f(z)=u(x,y)+\mathrm{i}v(x,y)$, then … ►If $f(z)$ is analytic in an open domain $D$, then each of its derivatives $f^{\prime }(z)$, $f^{\prime \prime }(z)$, $\mathrm{\dots }$ exists and is analytic in $D$. … ►exist. … ►If the limit exists, then the double series is convergent; otherwise it is divergent. …

30: 1.10 Functions of a Complex Variable

… ►An isolated singularity $z_0$ is always removable when ${lim}_{z\to z_0}f(z)$ exists, for example $(\sin z)/z$ at $z=0$. … ►Suppose $F(z)$ is multivalued and $a$ is a point such that there exists a branch of $F(z)$ in a cut neighborhood of $a$, but there does not exist a branch of $F(z)$ in any punctured neighborhood of $a$. … ►The last condition means that given $ϵ$ ($>0$) there exists a number $a_0\in [a,b)$ that is independent of $z$ and is such that …