4439.sagews

#integrate 1/z on the unit circle

f(z)=1/z # define the integrand

c(t)=e^(I*t)  # parameterize the unit circle c(t):[0,2pi]

start=0
stop=2*float(pi)
N=1000

line_points = srange(float(start), float(stop), (stop-start)/N,include_endpoint=True) 

z = map(c,line_points)

sum(f(z[i])*(z[i+1]-z[i]) for i in range(0,N-1)) # approximate integral of 1/z on the unit circle

-0.019719404719 + 6.27686082159*I

dc(t)=c.diff(t) # derivative of c(t) 
answer=integrate(f(c(t))*dc(t),(t,start,stop)) 
    # integrate using parametrization 
answer

6.28318530718*I

float(answer.real_part())+float(answer.imag_part())*I 
    # check if answer from parametrization agrees with approximation by taking its floating-point value

6.28318530718*I

# As we can see based on our solutions, the appoximation formula and parameterization is approximately close to I*2*float(pi), or also know as 6.283185*I