linear space of function
https://www.bilibili.com/video/BV1PX4y167RS
quantum state[48.2.3]
Taylor vs. Fourier[@ref(taylor-vs.-fourier)]
f(x)=a0x0+a1x1+a2x2+⋯=∞∑k=0akxk
f(x)=x0x0+x1x1+x2x2+⋯=∞∑k=0xkxk
Def: 48.3
⟨f|g⟩=∫ba¯¯¯¯¯¯¯¯¯¯¯f(x)g(x)dxf,g:R→R=∫baf(x)g(x)dx
Dirac bracket[48.5]
⟨x2∣∣x⟩x2,x:R→R=∫bax2xdx=∫bax3dx=[x44]ba≢0
x0⊥/x1,x1⊥/x2,⋯
⟨xm|xn⟩=∫baxmxndx=δmn
⟨1|xn⟩=∫bax0xndx=δ0n⇒x0=δ(x)=δ(x−0)
⟨xm|xn⟩=∫baxmxndx=δmn⇒xm=(−1)mm!δ(m)(x)
|f⟩=1|f⟩=(∑i∣∣^fi⟩⟨^fi∣∣)|f⟩=∑i∣∣^fi⟩⟨^fi∣∣f⟩
|f⟩=1|f⟩=(∑i∣∣^fi⟩⟨^fi∣∣)|f⟩=∑i∣∣^fi⟩⟨^fi∣∣f⟩=1|f⟩=(∑n|xn⟩⟨xn|)|f⟩=∑n|xn⟩⟨xn|f⟩=∑n⟨xn|f⟩|xn⟩⟨xn||f⟩=⟨xn|f⟩=∫baxnf(x)dx=∫ba(−1)nn!δ(n)(x)f(x)dx=f(n)(0)n!|f⟩=∑n⟨xn|f⟩|xn⟩=∑nf(n)(0)n!|xn⟩|f⟩=∑nf(n)(0)n!|xn⟩⇓f(x)=∑nf(n)(0)n!xn
beta function
https://www.bilibili.com/video/BV1pa4y1P7Da
(nk)=Cnk=n!(n−k)!k!=n(n−1)⋯(n−k+1)k!,{n∈Nk∈({0}∪N)(rk)=⎧⎨⎩r(r−1)⋯(r−k+1)k!k≥0,k∈Z0k<0,k∈Z
n∑k=0(rk)(⋅)
n∑k=−∞(rk)(⋅)=(0+0+⋯)+n∑k=0(rk)(⋅)
∞∑k=−∞(rk)(⋅)
n!=Γ(n+1)=∫∞0x(n+1)−1e−xdx
Γ(z)=∫∞0xz−1e−xdx
Γ(z+1)=zΓ(z)
Γ(z)Γ(1−z)=πsin(πz)
Γ(z)Γ(1−z)=πsin(πz)[Γ(z)Γ(1−z)]z=−n=[πsin(πz)]z=−n,n∈NΓ(−n)n!=Γ(n+1)=Γ(−n)Γ(1−(−n))=πsin(π(−n))=π−sin(nπ)Γ(−n)=−πn!sin(nπ)=−πn!0→−∞,n∈N
(nk)=Cnk=n!(n−k)!k!=Γ(n+1)Γ(n−k+1)Γ(k+1)
(nk)=Γ(n+1)Γ(n−k+1)Γ(k+1)
(nk)=Γ(n+1)Γ(n−k+1)Γ(k+1)k≤0=⎧⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪⎩Γ(n+1)Γ(n+1)Γ(1)=Γ(n+1)Γ(n+1)1=1k=0Γ(n+1)Γ(n−k+1)(−∞)=0k≤−1,k∈Z
beta function = β function
Definition 46.1 beta function = β function
B(p,q)=∫10xp−1(1−x)q−1dx=Γ(p)Γ(q)Γ(p+q)
(nk)=Γ(n+1)Γ(n−k+1)Γ(k+1)[(nk)]{n=a+bk=a=[Γ(n+1)Γ(n−k+1)Γ(k+1)]{n=a+bk=a(a+ba)=Γ(a+b+1)Γ(a+b−a+1)Γ(a+1)=Γ(a+b+1)Γ(b+1)Γ(a+1)
B(p,q)=Γ(p)Γ(q)Γ(p+q)[B(p,q)]{p=a+1q=b+1=[Γ(p)Γ(q)Γ(p+q)]{p=a+1q=b+1B(a+1,b+1)=Γ(a+1)Γ(b+1)Γ(a+1+b+1)=Γ(a+1)Γ(b+1)Γ([a+b+1]+1)=Γ(a+1)Γ(b+1)[a+b+1]Γ(a+b+1)
(a+ba)=Γ(a+b+1)Γ(b+1)Γ(a+1)=1Γ(b+1)Γ(a+1)Γ(a+b+1)=1[a+b+1]Γ(b+1)Γ(a+1)[a+b+1]Γ(a+b+1)=1[a+b+1]B(a+1,b+1)
https://en.wikipedia.org/wiki/Beta_function
https://en.wikipedia.org/wiki/Beta_function#Other_identities_and_formulas
https://en.wikipedia.org/wiki/Beta_function#Multivariate_beta_function
https://www.bilibili.com/video/BV1pa4y1P7Da/?t=4m