trigHypInts
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This content dictionary contains symbols which describe the functions which
perform the sine, cosine, hyperbolic sine and hyperbolic cosine integrals.
SinIntegral
The symbol SinIntegral represents the sine integral, it is
denoted by $Si(z)$.
The sine integral is defined by :
$Si(z)=\int_{0}^{z} \frac{sint}{t}dt$.
It is defined as in M. Abramowitz and I. Stegun, Handbook of
Mathematical Functions, [5.2.1]
Si : real $\rightarrow$ real
Si : complex $\rightarrow$ complex
Si : symbolic $\rightarrow$ symbolic
$Si(z)=\int_0^z \frac{sin(t)}{t}dt
$Si(z)=\sum_{n=0}^{\infty} \frac{(-1)^{n}z^{2n+1}}{(2n+1)(2n+1)!}$,
series expansion
$Si(-z)=-Si(z)$, symmetry
$Si(\bar{z})=\overline{Si(z)}$, symmetry
CosIntegral
The symbol CosIntegral represents the cosine integral, it is
denoted by $Ci(z)$.
The cosine integral is defined by
$Ci(z)=-\int_z^{\infty} \frac{cost}{t} dt$ where $|argz|<
\pi$.
It is defined as in M. Abramowitz and I. Stegun, Handbook of
Mathematical Functions,[5.2.27]
Ci : real $\rightarrow$ real
Ci : complex $\rightarrow$ complex
Ci : symbolic $\rightarrow$ symbolic
$Ci(z) = -\int_z^{\infty} \frac{cos t}{t} dt
$Ci(z)=\gamma + lnz + \sum_{n=1}^{\infty}
\frac{(-1)^{n}z^{2n}}{2n(2n)!}$, series expansion
symmetry relation: $Ci(\bar{z})=\overline{Ci(z)}$, symmetry
$Ci(-z)=Ci(z)-i \pi$, ($0 P argz P \frac{\pi}{2}$), symmetry
SinhIntegral
The symbol SinhIntegral represents the hyperbolic sine integral, it is denoted by
The hyperbolic sine integral is defined by $Shi(z)=\int_{0}^{z}
\frac{sinh(t)}{t} dt$.
It is defined as in M. Abramowitz and I. Stegun, Handbook of
Mathematical Functions,[5.2.3]
Shi : real $\rightarrow$ real
Shi : complex $\rightarrow$ complex
Shi : symbolic $\rightarrow$ symbolic
$Shi(z)=\int_{0}^{z} \frac{sinht}{t} dt$
$Shi(z)=\sum_{n=0}^{\infty}\frac{z^{2n+1}}
{(2n+1)(2n+1)!}$, series expansion
CoshIntegral
The symbol CoshIntegral represents the hyperbolic cosine
integral, it is denoted by $Chi(z)$.
The hyperbolic cosine integral is defined by $Chi(z)=\gamma
+ lnz + \int_{0}^{z} \frac{cosht-1}{t} dt$ where $\gamma=.5772156649
\dots$ is an Euler's constant and $|argz|P \pi$. It has a branch cut
along the negative real axis in the complex $z$ plane.
It is defined as in M. Abramowitz and I. Stegun, Handbook of
Mathematical Functions,[5.2.4]
Chi : real $\rightarrow$ real
Chi : complex $\rightarrow$ complex
Chi : symbolic $\rightarrow$ symbolic
$Chi(z)=\gamma + lnz + \int_{0}^{z} \frac{cosht-1}{t} dt~~~~
(|arg(z)| < /pi$
$Chi(z)=\gamma + lnz + \sum_{n=1}^{\infty} \frac{z^{2n}}{2n(2n)!}$,
series expansion