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11: 19.34 Mutual Inductance of Coaxial Circles

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12: 22.10 Maclaurin Series

… ►Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). …

13: 2.9 Difference Equations

… ► $c_0=1$, and higher coefficients are determined by formal substitution. … ►The coefficients $b_s$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when ${\alpha }_2-{\alpha }_1=0$, or with $b_0=1$ when ${\alpha }_2-{\alpha }_1=1,2,3,\mathrm{\dots }$. …

14: 19.28 Integrals of Elliptic Integrals

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15: 1.13 Differential Equations

… ►The substitution $\xi =1/z$ in (1.13.1) gives … ►The substitution … ►In (1.13.1) substitute …

16: 18.23 Hahn Class: Generating Functions

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17: 20.11 Generalizations and Analogs

… ►With the substitutions $a=q{\mathrm{e}}^{2\mathrm{i}z}$, $b=q{\mathrm{e}}^{-2\mathrm{i}z}$, with $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, we have …

18: 28.33 Physical Applications

… ►Substituting $z=\omega t$, $a=b/{\omega }^2$, and $2q=f/{\omega }^2$, we obtain Mathieu’s standard form (28.2.1). …

19: 10.41 Asymptotic Expansions for Large Order

… ►To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively. … ►Lastly, we substitute into (10.4.4), again with $z$ replaced by $\nu z$. …

20: 18.9 Recurrence Relations and Derivatives

… ►Further $n$-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7). … ►Further $n$-th derivative formulas relating two different Laguerre polynomials can be obtained from §13.3(ii) by substitution of (13.6.19). …