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21: 28.2 Definitions and Basic Properties

… ►Since (28.2.1) has no finite singularities its solutions are entire functions of $z$. Furthermore, a solution $w$ with given initial constant values of $w$ and $w^{\prime }$ at a point $z_0$ is an entire function of the three variables $z$, $a$, and $q$. … ► $\cos \left(\pi \nu \right)$ is an entire function of $a,q^2$. …

22: Mathematical Introduction

… ►This is because $𝐅$ is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as $𝐅$ is an entire function of each of its parameters $a$, $b$, and $c$:​ this results in fewer restrictions and simpler equations. …

23: 2.5 Mellin Transform Methods

… ►Furthermore, $ℳf_1\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane, whose singularities are simple poles at $-{\alpha }_s$, $s=0,1,2,\mathrm{\dots }$, with principal part … ►Similarly, if $\kappa =0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function on the entire $z$-plane with simple poles at ${\beta }_s$, $s=0,1,2,\mathrm{\dots }$, with principal part …Alternatively, if $\kappa \ne 0$ in (2.5.18), then $ℳh_2\left(z\right)$ can be continued analytically to an entire function. … ►Similarly, since $ℳh_2\left(z\right)$ can be continued analytically to a meromorphic function (when $\kappa =0$) or to an entire function (when $\kappa \ne 0$), we can choose $\rho$ so that $ℳh_2\left(z\right)$ has no poles in . …

24: 9.2 Differential Equation

… ►All solutions are entire functions of $z$. …

25: 15.2 Definitions and Analytical Properties

… ►The principal branch of $𝐅(a,b;c;z)$ is an entire function of $a$, $b$, and $c$. …

26: 16.5 Integral Representations and Integrals

… ►In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when . …

27: 20.2 Definitions and Periodic Properties

… ►For fixed $\tau$, each ${\theta }_j\left(z\right|\tau )$ is an entire function of $z$ with period $2\pi$; ${\theta }_1\left(z\right|\tau )$ is odd in $z$ and the others are even. …

28: 23.3 Differential Equations

… ►As functions of $g_2$ and $g_3$, $\mathrm{\wp }(z;g_2,g_3)$ and $\zeta (z;g_2,g_3)$ are meromorphic and $\sigma (z;g_2,g_3)$ is entire. …

29: 30.3 Eigenvalues

… ►With $\mu =m=0,1,2,\mathrm{\dots }$, the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form ${(1-x^2)}^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. …

30: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►If $b_1=b_2=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space. …