# Sines

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##### 2: 6.2 Definitions and Interrelations
###### §6.2(ii) Sine and Cosine Integrals
$\operatorname{Si}\left(z\right)$ is an odd entire function. …
##### 3: 4.35 Identities
4.35.16 $2\sinh u\cosh v=\sinh\left(u+v\right)+\sinh\left(u-v\right).$
4.35.17 ${\sinh}^{2}u-{\sinh}^{2}v=\sinh\left(u+v\right)\sinh\left(u-v\right),$
4.35.23 $\sinh\left(-z\right)=-\sinh z,$
4.35.29 $\sinh\left(3z\right)=3\sinh z+4{\sinh}^{3}z,$
4.35.34 $\sinh z=\sinh x\cos y+i\cosh x\sin y,$
##### 4: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\operatorname{Ein}\left(z\right)$; the logarithmic integral $\operatorname{li}\left(x\right)$; the sine integrals $\operatorname{Si}\left(z\right)$ and $\operatorname{si}\left(z\right)$; the cosine integrals $\operatorname{Ci}\left(z\right)$ and $\operatorname{Cin}\left(z\right)$.
##### 5: 6.17 Physical Applications
###### §6.17 Physical Applications
Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
##### 6: 10.64 Integral Representations
10.64.1 $\operatorname{ber}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\cos\left(x\sin t-nt\right)\cosh\left(x\sin t\right)\,\mathrm{d}t,$
10.64.2 $\operatorname{bei}_{n}\left(x\sqrt{2}\right)=\frac{(-1)^{n}}{\pi}\int_{0}^{\pi% }\sin\left(x\sin t-nt\right)\sinh\left(x\sin t\right)\,\mathrm{d}t.$
##### 7: 4.21 Identities
4.21.15 $2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v\right),$
4.21.17 $2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v\right).$
4.21.18 ${\sin}^{2}u-{\sin}^{2}v=\sin\left(u+v\right)\sin\left(u-v\right),$
4.21.24 $\sin\left(-z\right)=-\sin z,$
4.21.30 $\sin\left(3z\right)=3\sin z-4{\sin}^{3}z,$
##### 8: 4.1 Special Notation
 $k,m,n$ integers. …
The main functions treated in this chapter are the logarithm $\ln z$, $\operatorname{Ln}z$; the exponential $\exp z$, $e^{z}$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\operatorname{arcsin}z$, $\operatorname{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\operatorname{csch}z$, $\operatorname{sech}z$, $\coth z$; the inverse hyperbolic functions $\operatorname{arcsinh}z$, $\operatorname{Arcsinh}z$, etc. Sometimes in the literature the meanings of $\ln$ and $\operatorname{Ln}$ are interchanged; similarly for $\operatorname{arcsin}z$ and $\operatorname{Arcsin}z$, etc. …${\sin}^{-1}z$ for $\operatorname{arcsin}z$ and $\mathrm{Sin}^{-1}\;z$ for $\operatorname{Arcsin}z$.
##### 9: 4.14 Definitions and Periodicity
4.14.1 $\sin z=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2\mathrm{i}},$
4.14.5 $\csc z=\frac{1}{\sin z},$
The functions $\sin z$ and $\cos z$ are entire. In $\mathbb{C}$ the zeros of $\sin z$ are $z=k\pi$, $k\in\mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\tfrac{1}{2}\right)\pi$, $k\in\mathbb{Z}$. …
##### 10: 4.32 Inequalities
4.32.3 $|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y},$ $x>0$, $y>0$,