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##### 2: 28.21 Graphics
###### §28.21 Graphics Figure 28.21.1: Mc 0 ( 1 ) ⁡ ( x , h ) for 0 ≤ h ≤ 3 , 0 ≤ x ≤ 2 . Magnify 3D Help Figure 28.21.2: Mc 1 ( 1 ) ⁡ ( x , h ) for 0 ≤ h ≤ 3 , 0 ≤ x ≤ 2 . Magnify 3D Help Figure 28.21.3: Mc 0 ( 2 ) ⁡ ( x , h ) for 0.1 ≤ h ≤ 2 , 0 ≤ x ≤ 2 . Magnify 3D Help Figure 28.21.6: Ms 1 ( 2 ) ⁡ ( x , h ) for 0.2 ≤ h ≤ 2 , 0 ≤ x ≤ 2 . Magnify 3D Help
##### 3: 10.26 Graphics
###### §10.26(i) Real Order and Variable Figure 10.26.7: I ~ 1 / 2 ⁡ ( x ) , K ~ 1 / 2 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify Figure 10.26.8: I ~ 1 ⁡ ( x ) , K ~ 1 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify Figure 10.26.9: I ~ 5 ⁡ ( x ) , K ~ 5 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify Figure 10.26.10: K ~ 5 ⁡ ( x ) , 0.01 ≤ x ≤ 3 . Magnify
##### 4: 10.29 Recurrence Relations and Derivatives
###### §10.29(i) Recurrence Relations
With $\mathscr{Z}_{\nu}\left(z\right)$ defined as in §10.25(ii), …
$I_{0}'\left(z\right)=I_{1}\left(z\right),$
For results on modified quotients of the form $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ see Onoe (1955) and Onoe (1956).
###### §10.29(ii) Derivatives
They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
##### 6: 10.28 Wronskians and Cross-Products
###### §10.28 Wronskians and Cross-Products
10.28.1 $\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
##### 7: 28.1 Special Notation
and the modified Mathieu functions
 $\mathrm{Ce}_{\nu}\left(z,q\right)$, $\mathrm{Se}_{\nu}\left(z,q\right)$, $\mathrm{Fe}_{n}\left(z,q\right)$, $\mathrm{Ge}_{n}\left(z,q\right)$, $\mathrm{Me}_{\nu}\left(z,q\right)$, ${\mathrm{M}^{(j)}_{\nu}}\left(z,h\right)$, ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$, ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$, …
The functions ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ are also known as the radial Mathieu functions. … The radial functions ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ are denoted by ${\mathrm{Mc}^{(j)}_{n}}\left(z,q\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,q\right)$, respectively.
##### 8: 10.45 Functions of Imaginary Order
###### §10.45 Functions of Imaginary Order
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): … The corresponding result for $\widetilde{K}_{\nu}\left(x\right)$ is given by …
##### 9: 10.35 Generating Function and Associated Series
###### §10.35 Generating Function and Associated Series
10.35.4 $1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}\left(z\right)-2I_{6}\left(z% \right)+\dotsb,$
10.35.5 $e^{\pm z}=I_{0}\left(z\right)\pm 2I_{1}\left(z\right)+2I_{2}\left(z\right)\pm 2% I_{3}\left(z\right)+\dotsb,$
$\cosh z=I_{0}\left(z\right)+2I_{2}\left(z\right)+2I_{4}\left(z\right)+2I_{6}% \left(z\right)+\dots,$
$\sinh z=2I_{1}\left(z\right)+2I_{3}\left(z\right)+2I_{5}\left(z\right)+\dots.$
##### 10: 28.22 Connection Formulas
###### §28.22 Connection Formulas
The joining factors in the above formulas are given by …
28.22.13 ${\mathrm{M}^{(1)}_{\nu}}\left(z,h\right)=\frac{{\mathrm{M}^{(1)}_{\nu}}\left(0% ,h\right)}{\mathrm{me}_{\nu}\left(0,h^{2}\right)}\mathrm{Me}_{\nu}\left(z,h^{2% }\right).$
Here $\mathrm{me}_{\nu}\left(0,h^{2}\right)$ $(\neq 0)$ is given by (28.14.1) with $z=0$, and ${\mathrm{M}^{(1)}_{\nu}}\left(0,h\right)$ is given by (28.24.1) with $j=1$, $z=0$, and $n$ chosen so that $|c_{2n}^{\nu}(h^{2})|=\max(|c_{2\ell}^{\nu}(h^{2})|)$, where the maximum is taken over all integers $\ell$. …