[PYTHON] from sympy import symbols, integrate, exp, I, pi, Rational

Define the variable

theta = symbols('theta', real=True)

Define the path C: z = (5/2) * exp(i * theta) for theta from 0 to pi

r = Rational(5, 2) z = r * exp(I * theta) dz = z.diff(theta)

Define the function f(z) = z^2

f = z**2

Compute the integral over C

integral_C = integrate(f * dz, (theta, 0, pi))

print(f"Integral value: {integral_C}") [/PYTHON] ps_1 = """ Integral value: -125/12 """ [PYTHON] from math import gcd

numerator = 125 denominator = 12 common = gcd(numerator, denominator) print(f"GCD: {common}") print(f"p: {numerator // common}") print(f"q: {denominator // common}") print(f"p+q: {numerator // common + denominator // common}") [/PYTHON] ps_2 = """ GCD: 1 p: 125 q: 12 p+q: 137 """ The contour integral is given by

Since the function $f(z) = z^2$ is entire (analytic everywhere in the complex plane), the integral is path-independent and depends only on the endpoints of the contour $C$. The contour $C$ starts at $z_{start} = \frac{5}{2}$ and ends at $z_{end} = -\frac{5}{2}$.

An antiderivative of $f(z) = z^2$ is $F(z) = \frac{z^3}{3}$. By the Fundamental Theorem of Calculus for contour integrals, we have:

Calculating the values:

Substituting these back into the expression for $I$:

Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2:

We are given that $I = -\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Thus,

Checking for relative primality: $125 = 5^3$ and $12 = 2^2 \cdot 3$. They share no common factors, so they are relatively prime.

Finally, we find $p+q$:

$\boxed{137}$

The integral of $f(z) = z^2$ over the contour $C$ (the upper semicircle of radius $5/2$ traversed from $5/2$ to $-5/2$) can be computed using the fundamental theorem of calculus for analytic functions, as $z^2$ has the antiderivative $z^3/3$.

The Python code confirmed this result: $\int_C f(z)\, dz = -\frac{125}{12}$. Thus, $p = 125$ and $q = 12$. These are relatively prime since $125 = 5^3$ and $12 = 2^2 \cdot 3$. We need to find $p+q$:

$\boxed{137}$