Use FFT fft() to compute a Riemann sum - r

Question

I would like to calculate the Following Sum by applying Fast Fourier transforms (FFT). I want to compute the following Riemann sum approximation using FFT :
Here is the Psy function that I'm using :
Psyfun<-function(u,T,r,q,sigma,lmbda,meanV,v0,rho){
j <- as.complex(1i)
a <- lmbda*meanV
b <- lmbda
d <- sqrt((j*rho*sigma*u-b)**2+(u**2+j*u)*sigma**2)
g <- (b-j*rho*sigma*u-d)/(b-j*rho*sigma*u+d)
ret <- exp(j*u*(r-q)*T)
ret <- ret*exp((a/sigma**2)*((b - rho*j*sigma*u - d)*T - 2.0*log((1-g*exp(-d*T))/(1-g))))
return (ret*exp((v0/sigma**2)*(b - rho*j*sigma*u - d)*(1-exp(-d*T))/(1-g*exp(-d*T))))
}
Here are the sample parameters :
r = 0.025 ,q= 0.01, sigma = 0.2, lmbda = 0.5, meanV = 0.5, v0 = 0.5 , rho = 0.3
I want to compute the values for K and T equals to :
K1=172.77 and T1 = 0.197, K2= 75.63 and T2 = 0.563, K3 = 269.54 and T3 = 0.2648
I implement the following code to do it:
N=2^10 # Number of subdivision in [0,a]
alpha=2 # alpha
delta= 0.25 # delta= a/N where a is the up value of w (w in [0,a])
lambda=(2*pi)/(N*delta)
j=seq(1,N,1)
k=seq(1,N,1)
b=(lambda*N)/2
strike= -b+(k-1)*lambda
strike= exp(strike)
res=c()
for (i in 1:N){
w=delta*(i-1) # w= j*delta but from 1 to N so w=(i-1)*delta
w_FC=w-(alpha+1)*1i
phi= Psyfun(w_FC,T,r,q,sigma,lmbda,meanV,v0,rho)
phi=phi*exp(-r1*(T))
phi=phi/(alpha^2+alpha-w^2+1i*(2*alpha+1)*w)
phi=phi*delta*exp(1i*w*b)
res=rbind(res,phi)
}
Result=Re(fft(res))*exp(-alpha*(-b+(k-1)*lambda))/pi
I obtain k numbers of values so how to get the one that correspond to K1,K2,K3.
Can anybody recommend a procedure to implement the computation? Thanks
I have no previous experience in Fast Fourier transforms (FFT) processing, so I appreciate any tips and pointers related to the mathematics / methods / Code in addition to advice on how better to approach this programmatically.