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Staying with mpmath... Avter evaluating :

reset()
import mpmath as mp

mp.dps = 15

f = 8500
rho = 1.225
c0 = 343
Omega = 2*mp.pi*f
k = Omega/c0
Z = -426

def Integrand(x, y):
    # return a one-parameter anonymous function of kx at point x, y.
    # This function returns an mpc object
    return lambda kx: ((2*mp.sqrt(k**2 - kx**2)*Z)/(mp.sqrt(k**2 - kx**2)*Z + Omega*rho))*((mp.exp(1j*(kx*x + mp.sqrt(k**2 - kx**2)*y)))/(mp.sqrt(k**2 - kx**2)))

def integration(kx_interval, x, y):
    # builds the one-parameter integrand and integrates them.
    # Return the result as a Python complex
    itg = Integrand(x, y)
    rp = mp.quad(lambda kx:mp.re(itg(kx)), kx_interval)
    ip = mp.quad(lambda kx:mp.im(itg(kx)), kx_interval)
    return complex(rp + 1j*ip)

One gets :

sage: integration([-100*k, 100*k], 0, 0)
(356.37828188993564-16.743674967114078j)

Is that what you aimed at ?

Note that you do not need to separately integrate the real and imaginary parts :

# Shorter and cleaner
def integration1(kx_interval, x, y):
    # builds the one-parameter integrands and integrates them.
    # Return the result as a Python complex
    return complex(mp.quad(Integrand(x, y), kx_interval))

sage: integration1([-100*k, 100*k], 0, 0)
(356.37828188993564-16.743674967114078j)

HTH,

Staying with mpmath... Avter After evaluating :

reset()
import mpmath as mp

mp.dps = 15

f = 8500
rho = 1.225
c0 = 343
Omega = 2*mp.pi*f
k = Omega/c0
Z = -426

def Integrand(x, y):
    # return a one-parameter anonymous function of kx at point x, y.
    # This function returns an mpc object
    return lambda kx: ((2*mp.sqrt(k**2 - kx**2)*Z)/(mp.sqrt(k**2 - kx**2)*Z + Omega*rho))*((mp.exp(1j*(kx*x + mp.sqrt(k**2 - kx**2)*y)))/(mp.sqrt(k**2 - kx**2)))

def integration(kx_interval, x, y):
    # builds the one-parameter integrand and integrates them.
    # Return the result as a Python complex
    itg = Integrand(x, y)
    rp = mp.quad(lambda kx:mp.re(itg(kx)), kx_interval)
    ip = mp.quad(lambda kx:mp.im(itg(kx)), kx_interval)
    return complex(rp + 1j*ip)

One one gets :

sage: integration([-100*k, 100*k], 0, 0)
(356.37828188993564-16.743674967114078j)

Is that what you aimed at ?

Note that you do not need to separately integrate the real and imaginary parts :

# Shorter and cleaner
def integration1(kx_interval, x, y):
    # builds the one-parameter integrands and integrates them.
    # Return the result as a Python complex
    return complex(mp.quad(Integrand(x, y), kx_interval))

sage: integration1([-100*k, 100*k], 0, 0)
(356.37828188993564-16.743674967114078j)

HTH,

Staying with mpmath... After evaluating :

reset()
import mpmath as mp

mp.dps = 15

f = 8500
rho = 1.225
c0 = 343
Omega = 2*mp.pi*f
k = Omega/c0
Z = -426

def Integrand(x, y):
    # return a one-parameter anonymous function of kx at point x, y.
    # This function returns an mpc object
    return lambda kx: ((2*mp.sqrt(k**2 - kx**2)*Z)/(mp.sqrt(k**2 - kx**2)*Z + Omega*rho))*((mp.exp(1j*(kx*x + mp.sqrt(k**2 - kx**2)*y)))/(mp.sqrt(k**2 - kx**2)))

def integration(kx_interval, x, y):
    # builds the one-parameter integrand and integrates them.
    # Return the result as a Python complex
    itg = Integrand(x, y)
    rp = mp.quad(lambda kx:mp.re(itg(kx)), kx_interval)
    ip = mp.quad(lambda kx:mp.im(itg(kx)), kx_interval)
    return complex(rp + 1j*ip)

one gets :

sage: integration([-100*k, 100*k], 0, 0)
(356.37828188993564-16.743674967114078j)

Is that what you aimed at ?

Note that you do not need to separately integrate the real and imaginary parts :

# Shorter and cleaner
def integration1(kx_interval, x, y):
    # builds the one-parameter integrands and integrates them.
integrand and integrate it.
    # Return the result as a Python complex
    return complex(mp.quad(Integrand(x, y), kx_interval))

sage: integration1([-100*k, 100*k], 0, 0)
(356.37828188993564-16.743674967114078j)

HTH,