# Barrett’s

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##### 1: 7.20 Mathematical Applications
###### §7.20(ii) Cornu’s Spiral
Let the set $\{x(t),y(t),t\}$ be defined by $x(t)=C\left(t\right)$, $y(t)=S\left(t\right)$, $t\geq 0$. Then the set $\{x(t),y(t)\}$ is called Cornu’s spiral: it is the projection of the corkscrew on the $\{x,y\}$-plane. …
##### 2: 31.2 Differential Equations
###### §31.2(i) Heun’s Equation
31.2.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\frac{\epsilon}{z-a}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{% \alpha\beta z-q}{z(z-1)(z-a)}w=0,$ $\alpha+\beta+1=\gamma+\delta+\epsilon$.
##### 3: 29.2 Differential Equations
###### §29.2(ii) Other Forms
we have …For the Weierstrass function $\wp$ see §23.2(ii). …
##### 5: 7.2 Definitions
###### §7.2(ii) Dawson’s Integral
7.2.5 $F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\mathrm{d}t.$
7.2.8 $S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$
$\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection. …
$\lim_{x\to\infty}S\left(x\right)=\tfrac{1}{2}.$
##### 6: 28.20 Definitions and Basic Properties
###### §28.20(i) Modified Mathieu’s Equation
When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: For $s\in\mathbb{Z}$,
${\mathrm{M}^{(1)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)=e^{\mathrm{i}s\pi\nu}{% \mathrm{M}^{(1)}_{\nu}}\left(z,h\right),$
##### 8: 19.2 Definitions
Let $s^{2}(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. …Because $s^{2}$ is a polynomial, we have …
###### §19.2(iii) Bulirsch’s Integrals
Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …
##### 9: 28.8 Asymptotic Expansions for Large $q$
###### Barrett’s Expansions
Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …
##### 10: Bibliography B
• R. F. Barrett (1964) Tables of modified Struve functions of orders zero and unity.
• W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
• B. C. Berndt (1989) Ramanujan’s Notebooks. Part II. Springer-Verlag, New York.
• B. C. Berndt (1991) Ramanujan’s Notebooks. Part III. Springer-Verlag, Berlin-New York.
• M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.