# trigonometric form

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##### 2: 29.2 Differential Equations
29.2.4 $(1-k^{2}{\cos}^{2}\phi)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\phi}^{2}}+k^{2}% \cos\phi\sin\phi\frac{\mathrm{d}w}{\mathrm{d}\phi}+(h-\nu(\nu+1)k^{2}{\cos}^{2% }\phi)w=0,$
##### 3: 4.23 Inverse Trigonometric Functions
###### §4.23(iv) Logarithmic Forms
Care needs to be taken on the cuts, for example, if $0 then $1/(x+i0)=(1/x)-i0$. …
##### 5: 6.5 Further Interrelations
6.5.4 $\tfrac{1}{2}(\operatorname{Ei}\left(x\right)-E_{1}\left(x\right))=% \operatorname{Chi}\left(x\right)=\operatorname{Ci}\left(ix\right)-\tfrac{1}{2}% \pi i.$
##### 6: 6.7 Integral Representations
6.7.7 $\int_{0}^{1}\frac{e^{-at}\sin\left(bt\right)}{t}\,\mathrm{d}t=\Im\operatorname% {Ein}\left(a+ib\right),$ $a,b\in\mathbb{R}$,
6.7.8 $\int_{0}^{1}\frac{e^{-at}(1-\cos\left(bt\right))}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Ein}\left(a\right),$ $a,b\in\mathbb{R}$.
6.7.10 $\operatorname{Ein}\left(z\right)-\operatorname{Cin}\left(z\right)=\int_{0}^{% \pi/2}e^{-z\cos t}\sin\left(z\sin t\right)\,\mathrm{d}t,$
6.7.11 $\int_{0}^{1}\frac{(1-e^{-at})\cos\left(bt\right)}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Cin}\left(b\right),$ $a,b\in\mathbb{R}$.
##### 7: 6.2 Definitions and Interrelations
6.2.10 $\operatorname{si}\left(z\right)=-\int_{z}^{\infty}\frac{\sin t}{t}\,\mathrm{d}% t=\operatorname{Si}\left(z\right)-\tfrac{1}{2}\pi.$
##### 8: 28.2 Definitions and Basic Properties
With $\zeta={\sin}^{2}z$ we obtain the algebraic form of Mathieu’s equation …With $\zeta=\cos z$ we obtain another algebraic form: …
##### 9: 22.5 Special Values
###### §22.5(ii) Limiting Values of $k$
In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. …
##### 10: 4.45 Methods of Computation
The inverses $\operatorname{arcsinh}$, $\operatorname{arccosh}$, and $\operatorname{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. … The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). …