# at infinity

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##### 3: 31.11 Expansions in Series of Hypergeometric Functions
โบThe Fuchs-Frobenius solutions at $\infty$ are โบ
31.11.3_1 $P_{j}^{5}=\frac{{\left(\lambda\right)_{j}}{\left(1-\gamma+\lambda\right)_{j}}}% {{\left(1+\lambda-\mu\right)_{2j}}}z^{-\lambda-j}\*{{}_{2}F_{1}}\left({\lambda% +j,1-\gamma+\lambda+j\atop 1+\lambda-\mu+2j};\frac{1}{z}\right),$
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31.11.3_2 $P_{j}^{6}=\frac{{\left(\lambda-\mu\right)_{2j}}}{{\left(1-\mu\right)_{j}}{% \left(\gamma-\mu\right)_{j}}}z^{-\mu+j}\*{{}_{2}F_{1}}\left({\mu-j,1-\gamma+% \mu-j\atop 1-\lambda+\mu-2j};\frac{1}{z}\right).$
โบThen the Fuchs–Frobenius solution at $\infty$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with … โบFor example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\infty$. …
##### 4: 31.14 General Fuchsian Equation
โบThe general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_{j}$, $j=1,2,\dots,N$, and at $\infty$, is given by …The exponents at the finite singularities $a_{j}$ are $\{0,{1-\gamma_{j}}\}$ and those at $\infty$ are $\{\alpha,\beta\}$, where …
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##### 6: 31.12 Confluent Forms of Heun’s Equation
โบThis has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\infty$. … โบThis has irregular singularities at $z=0$ and $\infty$, each of rank $1$. … โบThis has a regular singularity at $z=0$, and an irregular singularity at $\infty$ of rank $2$. … โบThis has one singularity, an irregular singularity of rank $3$ at $z=\infty$. …
##### 7: 16.17 Definition
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• (ii)

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

• โบ
• (iii)

$L$ is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the $\Gamma\left(1-a_{\ell}+s\right)$ 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 $p=q\geq 1$.

• ##### 8: 16.5 Integral Representations and Integrals
โบSuppose 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. …
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##### 10: 15.11 Riemann’s Differential Equation
โบAlso, if any of $\alpha$, $\beta$, $\gamma$, is at infinity, then we take the corresponding limit in (15.11.1). …