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# 三角積分

## 定義

{\displaystyle {\begin{aligned}&\operatorname {Si} (z)=\int _{0}^{z}{\frac {\sin {t}}{t}}\mathrm {d} t\\&\operatorname {si} (z)=-\int _{z}^{\infty }{\frac {\sin {t}}{t}}\mathrm {d} t=\operatorname {Si} (z)-{\frac {\pi }{2}}\end{aligned}}}

${\displaystyle \operatorname {Ci} (z)=-\int _{z}^{\infty }{\frac {\cos {t}}{t}}\,\operatorname {d} \!t}$

{\displaystyle {\begin{aligned}\operatorname {Ci} (z)&=\gamma +\log {z}-\operatorname {Cin} (z)\\\operatorname {Cin} (z)&=\int _{0}^{z}{\frac {1-\cos {t}}{t}}\,\operatorname {d} \!t\end{aligned}}}

## 性質

### 微分積分

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

また、Si(z)のz→∞のときの値

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

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。

### 級数展開

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

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

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

### 指数積分との関係

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

## 参考文献

• Abramowitz, Milton [in 英語]; Stegun, Irene Ann [in 英語], eds. (1983) [June 1964]. "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 231. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253Mathar, R.J. (2009). "Numerical evaluation of the oscillatory integral over exp(iπxx1/x between 1 and ∞". Appendix B. arXiv:0912.3844 [math.CA]。
• Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). “Section 6.8.2 – Cosine and Sine Integrals”. Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8
• Sloughter, Dan. “Sine Integral Taylor series proof”. Difference Equations to Differential Equations. 2016年3月13日時点のオリジナルよりアーカイブ。2023年6月1日閲覧。
• Temme, N.M. (2010), “Exponential, Logarithmic, Sine, and Cosine Integrals”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255