4434.sagews

# integrate the function z on the broken line 0 to 1 to 1+i

f(z) = z #define the integrand

c1(t) = t #parametrize  with c1(t):[0,1] 
c2(t) = 1+I*t # parametrize with c2(t):[0,1]

start = 0
stop = 1
N = 1000    #c1(t) and c2(t) maps the interval [start, stop] onto the curve where f(z) is to be integrated. N is number of points to use in approximation of integral.

line_points = srange(float(start), float(stop), (stop-start)/N,include_endpoint=True)  
            #N+1 points on [start, stop]
z1 = map(c1,line_points)   
z2 = map(c2,line_points)                         
            #N+1 points on curve
            #The points are denoted z[0], z[1], ..., z[N]

X = sum(f(z1[i])*(z1[i+1]-z1[i]) for i in range(0,N-1))
Y = sum(f(z2[i])*(z2[i+1]-z2[i]) for i in range(0,N-1))
X+Y    # this is adding the approximate integral of z dz with the two integrands

0.999*I

dc1(t) = c1.diff(t) # derivative of c1(t)
dc2(t) = c2.diff(t) # derivative of c2(t)
answer1=integrate(f(c1(t))*dc1(t),(t,start,stop))
answer2=integrate(f(c2(t))*dc2(t),(t,start,stop))
answer = answer1+answer2  #We break down into two parametrizations and we take the integral of these separate parametrizations, and we then add them to get the solution

answer

I

float(answer.real_part())+float(answer.imag_part())*I

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_22.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZmxvYXQoYW5zd2VyLnJlYWxfcGFydCgpKStmbG9hdChhbnN3ZXIuaW1hZ19wYXJ0KCkpKkk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpWFjxXH/___code___.py", line 2, in <module> exec compile(u'float(answer.real_part())+float(answer.imag_part())*I' + '\n', '', 'single') File "", line 1, in <module> File "element.pyx", line 331, in sage.structure.element.Element.__getattr__ (sage/structure/element.c:2884) File "parent.pyx", line 327, in sage.structure.parent.getattr_from_other_class (sage/structure/parent.c:3236) AttributeError: 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic' object has no attribute 'real_part'

# Eventhough I'm unable to somehow find the solution for the previous cell, we see that the approximation sum and parametrization solution are closely similar to "I"