Example R programs and commands
		 2. Numerical integration; Fourier transform

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###### INTEGRATE A FUNCTION ALREADY KNOWN TO R
integrate(sin, 0,pi)     # interval is [0,pi]

###### INTEGRATE A USER-DEFINED FUNCTION
f <- function(x){return(exp(-x*x/2)/sqrt(2*pi) )}   # standard normal density
integrate(f, -1,1)
integrate(f, -1.96,1.96)

###### DISCRETE FOURIER TRANSFORM
N<-1024                 # Large for visualization, power of 2 for speed
t<- 2*pi*(0:(N-1))/N    # N points in [0,2*pi)
plot(t,sin(t),type="l") # Trigonometric function plot
matplot(t,cbind(cos(t),sin(t)), type="l")   # Two plots on same axes

# Make a vector of complex numbers:
freq<-5; f<-complex(N, real=cos(freq*t), imaginary=sin(freq*t))
matplot(t,cbind(Re(f),Im(f)), type="l")   # ...and plot Re and Im parts

Tf <- fft(f)            # Discrete ("fast") Fourier transform
matplot(cbind(Re(Tf),Im(Tf)), type="l")  # ...plot Re and Im parts of FFT

# Another example of the FFT:
freq<-50; f<-complex(N, real=cos(freq*t), imaginary=sin(freq*t))
matplot(t,cbind(Re(f),Im(f)), type="l")
Tf <- fft(f)
matplot(cbind(Re(Tf),Im(Tf)), type="l")