trigHypInts http://www.openmath.org/CDs/trigHypInts.ocd 1/1/5000 private alg1 arith1 calculus1 complex1 fns1 interval1 logic1 nums1 relation1 transc1 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