三角積分

• ${\displaystyle {d \over dz}\operatorname {Si} (z)={d \over dz}\operatorname {si} (z)={\frac {\sin(z)}{z}}}$
• ${\displaystyle {d \over dz}\operatorname {Ci} (z)={\frac {\cos(z)}{z}}}$
• ${\displaystyle \int \operatorname {Si} (z)dz=z\operatorname {Si} (z)+\cos(z)+C}$
• ${\displaystyle \int \operatorname {Ci} (z)dz=z\operatorname {Ci} (z)-\sin(z)+C}$

また、Si(z)のz→∞のときの値

${\displaystyle \lim _{z\rightarrow \infty }\operatorname {Si} (z)=\int _{0}^{\infty }{\frac {\sin(t)}{t}}dt={\frac {\pi }{2}}}$

はディリクレ積分といい、複素積分などを用いることによって示せる。

ローラン級数

${\displaystyle \operatorname {Si} (z)=z\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{(2k+1)^{2}(2k)!}}}$

${\displaystyle \operatorname {Ci} (z)=\gamma +\log(z)+{\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{k(2k)!}}}$

ベッセル級数

${\displaystyle \operatorname {Si} (z)=\pi \sum _{k=0}^{\infty }J_{{\frac {1}{2}}+k}{\Bigl (}{\frac {z}{2}}{\Bigr )}^{2}}$

超幾何級数

${\displaystyle \operatorname {Si} (z)=z\cdot {_{2}F_{1}}\left[{\begin{matrix}{\frac {1}{2}}\\{\frac {3}{2}},{\frac {3}{2}}\end{matrix}};-{\frac {z^{2}}{4}}\right]}$

${\displaystyle {\begin{aligned}\operatorname {Ci} (z)&=\gamma +\log {z}-{\frac {z^{2}}{4}}\cdot {_{2}F_{3}}\left[{\begin{matrix}1,1\\2,2,{\frac {3}{2}}\end{matrix}};-{\frac {z^{2}}{4}}\right]\\\end{aligned}}}$

${\displaystyle \operatorname {Ein} (\pm iz)=\operatorname {Cin} (z)\pm i\operatorname {Si} (z)}$