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1: 13.12 Products

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2: 7.2 Definitions

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Values at Infinity

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Values at Infinity

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3: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by …The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …

4: 6.2 Definitions and Interrelations

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Values at Infinity

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5: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►This has irregular singularities at $z=0$ and $\mathrm{\infty }$, each of rank $1$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. … ►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty }$. …

6: 16.17 Definition

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$L$ is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all $z$ ($\ne 0$) if , and for if $p=q\ge 1$.

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$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$ once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all $z$ if $p>q$, and for $|z|>1$ if $p=q\ge 1$.

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7: 1.13 Differential Equations

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Transformation of the Point at Infinity

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8: 31.11 Expansions in Series of Hypergeometric Functions

… ►Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … ►For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. …

9: 16.5 Integral Representations and Integrals

… ►Suppose first that $L$ is a contour that starts at infinity on a line parallel to the positive real axis, encircles the nonnegative integers in the negative sense, and ends at infinity on another line parallel to the positive real axis. …

10: 15.11 Riemann’s Differential Equation

… ►Also, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). …