# terminant function

(0.001 seconds)

## 11—20 of 30 matching pages

##### 11: 16.4 Argument Unity
See Erdélyi et al. (1953a, §4.4(4)) for a non-terminating balanced identity. … when $\Re\left(2c-a-b\right)>-1$, or when the series terminates with $a=-n$. … when $\Re\left(b+c+d-a\right)<1$, or when the series terminates with $d=-n$. … when the series on the right terminates and the series on the left converges. When the series on the right does not terminate, a second term appears. …
##### 12: 35.8 Generalized Hypergeometric Functions of Matrix Argument
###### §35.8(i) Definition
If $-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}$ for some $j,k$ satisfying $1\leq j\leq p$, $1\leq k\leq m$, then the series expansion (35.8.1) terminates. … If $p>q+1$, then (35.8.1) diverges unless it terminates.
##### 14: 5.11 Asymptotic Expansions
###### §5.11 Asymptotic Expansions
and … If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\geq 0$) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. …
##### 15: 34.2 Definition: $\mathit{3j}$ Symbol
34.2.6 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},$
where ${{}_{3}F_{2}}$ is defined as in §16.2. For alternative expressions for the $\mathit{3j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{3}F_{2}}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
##### 16: 34.4 Definition: $\mathit{6j}$ Symbol
34.4.3 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),$
where ${{}_{4}F_{3}}$ is defined as in §16.2. For alternative expressions for the $\mathit{6j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{4}F_{3}}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).
##### 17: 10.61 Definitions and Basic Properties
###### §10.61(iii) Reflection Formulas for Arguments
In general, Kelvin functions have a branch point at $x=0$ and functions with arguments $xe^{\pm\pi i}$ are complex. …
##### 18: 17.9 Further Transformations of ${{}_{r+1}\phi_{r}}$ Functions
17.9.3 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(abz/c;q\right)_{% \infty}}{\left(bz/c;q\right)_{\infty}}{{}_{3}\phi_{2}}\left({a,c/b,0\atop c,cq% /(bz)};q,q\right)+\frac{\left(a,bz,c/b;q\right)_{\infty}}{\left(c,z,c/(bz);q% \right)_{\infty}}{{}_{3}\phi_{2}}\left({z,abz/c,0\atop bz,bzq/c};q,q\right),$
##### 19: 6.12 Asymptotic Expansions
If the expansion is terminated at the $n$th term, then the remainder term is bounded by $1+\chi(n+1)$ times the next term. For the function $\chi$ see §9.7(i). …