How to integrate $\int_0^T K_{\lambda}(t,s) \sigma(s)ds$ using scipy.integrate

Question

I need to integrate the expression

using scipy's integrate module.

I have tried

def K(t,s):
    return np.exp(lbda*(T-s))*np.exp(lbda*t)/(1-np.exp(lbda*T))

def sigma(s):
    return np.sin(2*np.pi*s)

def integral(t):
    result = integrate.quad(K*sigma, 0, T)
    return  result

But obviously it's wrong since K has two variables, one of which would be undefined. How do I proceed?

For reference the function integral would be called with different values for t.

Answer 1

The first parameter of quad must be a callable whose first parameter is the integration variable. You'll have to create a new function (or use a lambda expression) for the integrand that calls your existing functions. It is also probably a good idea to make all the parameters that you'll use into function arguments, to avoid the need for global variables. You can pass these in via the args parameter of quad.

For example,

def K(t, s, lbda, T):
    return np.exp(lbda*(T-s))*np.exp(lbda*t)/(1-np.exp(lbda*T))

def sigma(s):
    return np.sin(2*np.pi*s)

def integrand(s, t, lbda, T):
    return K(t, s, lbda, T) * sigma(s)

def integral(t, lbda, T):
    result = integrate.quad(integrand, 0, T, args=(t, lbda, T))[0]
    return result

Then the code that uses these function might look something like this:

# Define the parameters
lbda = 1.25
T = 8.0

# Define the t value.
t = 0.5

# Evaluate the integral at t.
y = integral(t, lbda, T)