TeXiS: Una Plantilla De LaTeX Para Tesis Y Otros Os Manual Te Xi S

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ζ
∞
F1
1α= 1 L= 1
ζA(s)
A
Aφ(x) = −∂2
xφ(x)
φ(0) = 0
∂xφ(L) + γφ(L)=0
λ > 0
λ
γCos(Lλ) + Sin(Lλ)=0
λ
γ+T g(λL)=0
ζ
ζA(s) =
∞
X
n=1
λ−s
n
∞
ζA(s)
µ=λL θ =γL
T g[µ] + µ
θ= 0
µ
θCos[µ] + Sin[µ] = 0
µn
T g(µ)µn
µn
T g(x)
∞
µn=nπ +π
2+n
Donde n→0si cuando n →0
n
n
n→ ∞
→0
Sin(nπ +π
2+n) = −µn
θCos(nπ +π
2+n)
∞
X
p=0
(−1)n2p
(2p)! =−1
θ(nπ +π
2+)
∞
X
p=0
(−1)n2p+1
(2p+ 1)!
1 =
∞
X
p=1
2p+2 (−1)p1
(2p+ 2)! +1
θ
(1)
(2p+ 1)!+1
θnπ +π
2∞
X
p=0
(−1)p2p+1
(2p+ 1)!
n
=(1) +(2) +(3) +...
(1) =θ
nπ
(2) =−θ
2πn2
(3) =θ
n3π1
4−θ2
2π2−θ
π2
µnλn=µn
L
λn=αn +β+γ
n+δ
n2+η
n3+O(1
n4)
α, β, ... (1), (2), ...
ζA(s)
ζA(s) = P∞
n=1 λ−s
n=P∞
n=1 αn +β+γ
n+δ
n2+η
n3+O(1
n4)−2s=
P∞
n=1(αn)−2s
1 + β
αn +γ
αn2+δ
αn3+η
αn4+O(1
n5)
| {z }
χn
−2s
χn→0
ζA(s) =
∞
X
n=1
(αn)−2S(circulo)0 χ0+ (circulo)1 χ1+ (circulo)2 χ2+...
χn
1
n
(circulo)p
P∞
n=1(αn)−2s
1−sχn+s
2(s+ 1)χ2
n+−1
6s(2 + 3s+s2)χ3
n+s
24 (6 + 11s+ 6s2+s3)χ4
n
∞
X
n=1
(αn)−2S(circulo)0 =
∞
X
n=1
(αn)−2s=ζ(2s)
P∞
n=1(αn)−2S(circulo)1 χ1=
P∞
n=1(−sχn) =
−sα−sβ
αζ(s+ 1) + γ
αζ(s+ 2) + δ
αζ(s+ 3) + η
αζ(s+ 4) + PO(1
ns−5
P∞
n=1 (αn)−2S(circulo)2 = Ps
2(s+ 1)χ2
n
s(s+1)
2α−2sβ2
α2ζ(s+ 2) + 2βγ
αζ(s+ 3) + 2βγ
α2+γ2
α2)ζ(s+ 4)+PO(1
n5)
P∞
n=1(αn)−2S(circulo)3 =
P(αn)−s(−s
6(2 + 3s+s2))χ3
n
−α−2ss
6(2 + 3s+s2)(β
α)3ζ(s+ 3) + 3β2γ
α3ζ(s+ 4) + P1
ns−5
P(αn)2ss
24 (6 + 11s+ 6s2+s3)χ4
n
(circulo)4 = α−ss
24 (6 + 11s+ 6s2+s3)β
αζ(s+ 4)
ζ(2s+n)
ζA(s) = Csζ(s)+Cs+1 ζ(s+1)+Cs+2 ζ(s+2)+Cs+3 ζ(s+3)+Cs+4 ζ(s+4)+....
ζ(s+n)Cs+n
(circulo)n α, β, γ, ...
ζ
ζ
ζ
ζA(s) = 1
2πi RC
f0(x)
f(x)z−2s=
1
2πi RC
cos(γz)(γ+1
L)−sin(γz)zγ
L
cos(γz)z
L+sin(γz)z−2sdz
1
πSin(πs)Z∞
1
t−2s(1 + Lγ)Cosh(γt) + tγSin(γt)
t Cosh(tγ) + LSinh(tγ)
| {z }
χ
dt
χ
γ+1
t−L
t2+L2
t3+O(1
t4)
ζA(s) = Sin(πs)
πγ
2s−1+1
2s−L
2s+ 1 +L2
2s+ 2 +...
Aφ(x) = −∂2
xφ(x)−α
xφ(x)
α > 0
φ(0) = φ(L)=0
A[φ(x)] = ω2φ(x)
ω∈R
y1y2
φ(x) = C[1] e−iωx x F 1
1(1 + iα
2ω,2,2iωx)
| {z }
y1
+C[2] e−iωx x U(1 + iα
2ω,2,2iωx)
| {z }
y2
F1
1α= 1 L= 1
F1
1(a, b, z)U(a, b, z)
z ∂2
zψ(a, b, z)+(b−z)∂zψ(a, b, z)−a ψ(a, b, z) = 0
φ(0) = 0 U(1+ iα
2ω,2,0) →
∞C[2] = 0
y1(L) = 0
F1
1(1 + iα
2ω,2,2iωx)
F1
1
ω α = 1, L = 1
|z|→∞
F1
1(a, b, z) = Γ(b)ezza−b
Γ(a)∗A1+(−z)−a
Γ(b−a)∗A2
A1A2
µ=ωL β =αL
F1
1(1 + iβ
2µ,2,2iβ) = e−π
4
β
µ
2µ ei(β
2mu Ln(2µ)−π/2+2µ))
Γ(1 + iβ
2µ)+e−i(β
2mu Ln(2µ)−π/2))
Γ(1 −iβ
2µ)!
M(µ) = ei(β
2µLn(2µ)−π/2+2µ))
Γ(1 + iβ
2µ)+e−i(β
2µLn(2µ)−π/2))
Γ(1 −iβ
2µ)
ζA(s)
ζA(s)
ζA(s)
f(ω)
f0(ω)f(ω)ω±it
q=α
2βLog[2ωL]
q0=∂ωq
p= 2iµ
p0=∂ωp
Γ±= Γ(1 ±iα
2ω)
ψ±=1
Γ±∂ωΓ±
f(ω) = −iei(q+p)
Γ+
+ie−i q
Γ−
∂ωf(ω)
f0=eip eiq 1
Γ+
∂ω(q+p) + i∂ωΓ+
Γ2
++e−iq 1
Γ−
∂ωq−i∂ωΓ−
Γ2
−
| {z }
∗
∗ω=±it
∀t
eip
eip f y f0
f0
f=1
iq0−i∂ωΓ−
Γ−
P ara ω =it
f0
f=iq0+p0+i∂ωΓ+
Γ+
P ara ω =−it
qyp
α
2iω21−Log(2ωL) + ψ(1 −α
2βω )
iα
2ω21−Log(2ωL) + ψ(1 + α
2βω )+ 2iL
ζA(s) =
1
2πi R1
∞
iα
2t21−Log(2Lt)−iπ
2+ψ(1 −α
2t)t−2se−iπs (idt)+
1
2πi R∞
1
α
2it21−Log(2Lt)−iπ
2+ψ(1 −α
2t)+ 2iLt−2seiπs(−idt)
2iL t
1
2πi Z∞
1
2iLeiπst−2s(−idt) = Leiπs
2πi
1
s−1/2
s= 1/2
Res(s= 1/2) = L
2π
−α
2πsin[πs]R∞
1t−2s−s1−Log[2Lt] + ψ(1 −α
2t)dt −α
4Cos[πs]R∞
1t−2s−sdt
ψ ψ t → ∞
ψ(1 −α
2t)≈ −γ+O(1
t)
zetaA(s)
ζ(s)A=Leiπs
2πi
1
s−1/2−αSin[πs]
4π
1
s+1/2
α1+SLog(4)+2SLog[L]+Log[2L]
8π
sin(πs)
(s+1/2)2
γαSin[πs]
4π
1
s+1/2
αcos(πs)
8π
1
s+1/2
psi
s=−3/2
s=−1/2
−α
8π
1
(s+ 1/2)2+α1−γ−Log[2L]
4π
1
s+ 1/2+Regular
s=−1/2
Log[2L]
R∞
1Log[2Lω]dω
E0=~
2ζ(s)|−1
2
ζ(s)s=−1/2
(Ec)2s+1 s=−1/2
Ec≈1+2Log[Ec](s+ 1/2) + 2Log[Ec]2(s+ 1/2)2+O(s+ 1/2)3
E0=1
(s+ 1/2)2−α
8π+α(1 −γ−Log[2L]) −αLog[Ec]
4π(s+ 1/2) +Regular|s=−1/2
¡¡TODO!!
2
2
2
•
•
ε
2

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