# as functions of parameters

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##### 1: 31.13 Asymptotic Approximations
###### §31.13 Asymptotic Approximations
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …
##### 3: 28.21 Graphics
###### §28.21 Graphics Figure 28.21.6: Ms 1 ( 2 ) ⁡ ( x , h ) for 0.2 ≤ h ≤ 2 , 0 ≤ x ≤ 2 . Magnify 3D Help
##### 4: 8.17 Incomplete Beta Functions
8.17.4 $I_{x}\left(a,b\right)=1-I_{1-x}\left(b,a\right).$
8.17.6 $I_{x}\left(a,a\right)=\tfrac{1}{2}I_{4x(1-x)}\left(a,\tfrac{1}{2}\right),$ $0\leq x\leq\frac{1}{2}$.
8.17.13 $(a+b)I_{x}\left(a,b\right)=aI_{x}\left(a+1,b\right)+bI_{x}\left(a,b+1\right),$
8.17.20 $I_{x}\left(a,b\right)=I_{x}\left(a+1,b\right)+\frac{x^{a}(x^{\prime})^{b}}{a% \mathrm{B}\left(a,b\right)},$
8.17.21 $I_{x}\left(a,b\right)=I_{x}\left(a,b+1\right)-\frac{x^{a}(x^{\prime})^{b}}{b% \mathrm{B}\left(a,b\right)}.$
##### 5: 16.22 Asymptotic Expansions
For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).
##### 6: 33.13 Complex Variable and Parameters
###### §33.13 Complex Variable and Parameters
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 8: 20.1 Special Notation
 $m$, $n$ integers. … the nome, $q=e^{i\pi\tau}$, $0<\left|q\right|<1$. Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$, $\Im\tau>0$, so that $0 and $\tau$ and $q$ are uniquely related. …
##### 9: Bibliography U
• F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
• ##### 10: 31.1 Special Notation
Sometimes the parameters are suppressed.