| 誤差関数
erf(x), erf(z) |
|
∫exp (-x2)dx=((√π)/2)erf(x) ∫exp (x2)dx=-i ((√π)/2)erf (i x) |
| ∫0∞ exp (-x2)dx=(√π)/2 |
| 正弦積分関数Si
(x) |
余弦積分関数Ci
(x) |
|
∫sin (x)/x dx=Si (x) |
∫cos (x)/x dx=Ci (x) |
| Polylogarithm
function Lin(x)(ポリログ関数), 多重対数関数1)
[特殊関数] Lin+1(z)=∫0z Lin(t)/t dt, Li1(z)=-ln(1-z) n=2の時dilogarithm function Li2(z)(2重対数関数1),Spence's function) Li2(z)=-∫0z {log(1-z)}/z dz n=3の時trilogarithm function(3重対数関数) |
| デバイ模型1),
デバイ比熱式2),3)に
現れる積分 ∫[0,x] x4 ex/(ex-1)2 dx=12x2 Li2(ex)-24x Li3(ex)+24 Li4(ex)-x4 +4x3 ln(1-ex)-(x4/(ex-1)) ∫[0,x] x3/(ex-1) dx=6Li4(ex)-6x Li3(ex)+3x2 Li2(ex)+x3 ln(1-ex) -x4/4 |
| ∫0∞ x4
ex/(ex-1)2
dx=Γ(5)ζ(4)=(4!)π4/90=(4/15)π4≒25.975757609067316596384088716999 ∫0∞ x3/(ex-1) dx =π4/15≒6.493939402266829149096022 |
| Fresnel正弦積分: S(z)=∫0xsin(πt2/2) dt | Fresnel余弦積分: C(x)=∫0xcos(πt2/2)
dt |
| ∫sin(x2/2)dx=√πS(x/√π) |
∫cos(x2/2)dx=(√π) C(x/√π) |
| 指数積分関数Ei
(x), Ei (z) |
|
|
∫exp (x)/x dx=-Ei (i, -x) |
∫exp (-x)/x dx=Ei (i, -x) |
| Lambert's
W function ランベルトW関数1)
z=W(z)eW(z) のW(z) y=W(x)のグラフ1) |
W(x)計算サイト1)
|
| W(-π/2)=iπ/2 W(-1/e)=-1 W(-ln(2)/2)=-ln(2) W(0)=0 W(e)=1 W(1)=Ω(オメガ定数) |
∫W(x)dx=x{W(x)-1+(1/W(x))}+C |
| xx=zの解：x=ln(z)/W(ln(z))=exp(W(ln(z))) |
| ガンマ関数1)
Γ(x), Γ(k,
x) 別呼称:Omega function, product log(対数積) nが自然数のとき Γ(n+1)=n! Γ(1)=0!=1,Γ(1/2)=√π Γ(z)=∫0∞ tz-1 e-t dt (Re z >0) 留数Res(Γ(z),-n)=(-1)n/n! |
ホワイタッカーのM 関数Whittaker M (m,n,x) |
| ∫exp(x3) dx | =(x/9){-3Γ(1/3,-x3)
Γ(2/3)+2π√3}/{Γ(2/3) (-x3)1/3} |
| ∫exp(-x3) dx | ={x exp(-x3)/4}{3exp(x3/2) M (1/6, 2/3, x3)+4(x3)1/6}/{(x3)1/6} |
| ∫0∞exp (-x3) dx=2π√3/{9Γ(2/3)} | ∫0∞exp (-x5) dx=π/{5sin (π/5)Γ(4/5)} |
