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## Copyright 2021-2023 Ian Jauslin
##
## Licensed under the Apache License, Version 2.0 (the "License");
## you may not use this file except in compliance with the License.
## You may obtain a copy of the License at
##
## http://www.apache.org/licenses/LICENSE-2.0
##
## Unless required by applicable law or agreed to in writing, software
## distributed under the License is distributed on an "AS IS" BASIS,
## WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
## See the License for the specific language governing permissions and
## limitations under the License.
# approximate \int_a^b f using Gauss-Legendre quadratures
@everywhere function integrate_legendre(
f::Function,
a::Float64,
b::Float64,
weights::Tuple{Array{Float64,1},Array{Float64,1}}
)
out=0
for i in 1:length(weights[1])
out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)
end
return out
end
# \int f*g where g is sampled at the Legendre nodes
@everywhere function integrate_legendre_sampled(
f::Function,
g::Array{Float64,1},
a::Float64,
b::Float64,
weights::Tuple{Array{Float64,1},Array{Float64,1}}
)
out=0
for i in 1:length(weights[1])
out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)*g[i]
end
return out
end
# approximate \int_a^b f/sqrt((b-x)(x-a)) using Gauss-Chebyshev quadratures
@everywhere function integrate_chebyshev(
f::Function,
a::Float64,
b::Float64,
N::Int64
)
out=0
for i in 1:N
out=out+pi/N*f((b-a)/2*cos((2*i-1)/(2*N)*pi)+(b+a)/2)
end
return out
end
# approximate \int_0^\infty dr f(r)*exp(-a*r) using Gauss-Chebyshev quadratures
@everywhere function integrate_laguerre(
f::Function,
a::Float64,
weights_gL::Tuple{Array{Float64,1},Array{Float64,1}}
)
out=0.
for i in 1:length(weights_gL[1])
out+=1/a*f(weights_gL[1][i]/a)*weights_gL[2][i]
end
return out
end
# Hann window
@everywhere function hann(
x::Float64,
L::Float64
)
if abs(x)<L/2
return cos(pi*x/L)^2
else
return 0.
end
end
# Fourier transform (in 3d)
@everywhere function hann_fourier(
k::Float64,
L::Float64
)
return L^2*4*pi^3/k*(((k*L)^3-4*k*L*pi^2)*cos(k*L/2)-2*(3*(k*L)^2-4*pi^2)*sin(k*L/2))/((k*L)^3-4*k*L*pi^2)^2
end
# normalized Gaussian (in 3d)
@everywhere function gaussian(
k::Float64,
L::Float64
)
return exp(-k^2/(2*L))/sqrt(8*pi^3*L^3)
end
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