In [1]:

    

from sympy import init_printing; init_printing(use_latex='mathjax')
import sympy as sym


    

In [6]:

    

r, theta, k  = sym.symbols('r theta k')
r, theta, k


    

    
Out[6]:





$$\left ( r, \quad \theta, \quad k\right )$$

In [7]:

    

n = sym.Symbol('n', positive = True, integer=True)
n


    

    
Out[7]:





$$n$$

In [8]:

    

def lamb(n,k):
    return sym.Symbol('lambda_%s%s'%(n,k), positive = True)


    

In [10]:

    

lamb(0,k)


    

    
Out[10]:





$$\lambda_{0k}$$

In [11]:

    

f = 1 - r**2; f


    

    
Out[11]:





$$- r^{2} + 1$$

In [13]:

    

integrand = f * sym.besselj(n, lamb(n,k) * r) * sym.cos(n *theta) * r
integrand


    

    
Out[13]:





$$r \left(- r^{2} + 1\right) \cos{\left (n \theta \right )} J_{n}\left(\lambda_{nk} r\right)$$

In [14]:

    

ank = sym.Integral(integrand, (r, 0, 1), (theta, 0, 2*sym.pi))
ank


    

    
Out[14]:





$$\int_{0}^{2 \pi}\int_{0}^{1} r \left(- r^{2} + 1\right) \cos{\left (n \theta \right )} J_{n}\left(\lambda_{nk} r\right)\, dr\, d\theta$$

In [15]:

    

solution = ank.doit()
solution


    

    
Out[15]:





$$0$$

In [16]:

    

integ = lambda n: f * sym.besselj(n, lamb(n,k) * r) * sym.cos(n*theta) * r


    

In [17]:

    

integ(0)


    

    
Out[17]:





$$r \left(- r^{2} + 1\right) J_{0}\left(\lambda_{0k} r\right)$$

In [18]:

    

a0k = sym.Integral(integ(0), (r, 0, 1), (theta, 0, 2*sym.pi))
a0k


    

    
Out[18]:





$$\int_{0}^{2 \pi}\int_{0}^{1} r \left(- r^{2} + 1\right) J_{0}\left(\lambda_{0k} r\right)\, dr\, d\theta$$

In [19]:

    

a0k.doit()


    

    
Out[19]:





$$2 \pi \left(- \frac{2 J_{0}\left(\lambda_{0k}\right)}{\lambda_{0k}^{2}} + \frac{4 J_{1}\left(\lambda_{0k}\right)}{\lambda_{0k}^{3}}\right)$$

In [ ]: