Let
and consider the upper semicircular arc
Let
and form the closed contour
where is traversed from to , and is traversed from to .\
Suppose the contour integral
where and are positive integers which are relatively prime. Find .
[THOUGHT] To compute the contour integral ∫_C f(z) dz where f(z) = z² and C is the upper semicircular arc from z = 5/2 to z = -5/2, I will:
- Parameterize C: z = (5/2) e^{iθ}, dz = (5/2) i e^{iθ} dθ, θ from 0 to π.
- Substitute into the integral: ∫_C z² dz = ∫_0^π [(5/2)² e^{i2θ}] * [(5/2) i e^{iθ}] dθ.
- Simplify the integrand and compute the integral using sympy.
- Verify by computing the integral over the line segment L and using the fact that the closed contour integral is zero (since f is entire).
- Confirm that the result is -125/12, so p = 125, q = 12, and p+q = 137. [/THOUGHT] [PYTHON] import sympy as sp
Define variable
theta = sp.symbols('theta', real=True)
Parameterization of C
z = (5/2) * sp.exp(sp.I * theta) dz = (5/2) * sp.I * sp.exp(sp.I * theta)
Integrand
integrand_C = z**2 * dz
Compute integral over C
integral_C = sp.integrate(integrand_C, (theta, 0, sp.pi)) print(integral_C)
Alternatively, compute integral over L and take negative
x = sp.symbols('x', real=True) integrand_L = x**2 integral_L = sp.integrate(integrand_L, (x, -5/2, 5/2)) integral_C_alt = -integral_L print(integral_C_alt) [/PYTHON]
[THOUGHT] The Python output shows the integral value as approximately -10.4166666666667, which equals -125/12. Thus, p=125 and q=12 are relatively prime, so p+q=137. [/THOUGHT]