An example using SymPy.stats to compute the characteristic function of a general Cauchy random variable. This is run in isympy (hence the pretty printing)

t-characteristic

In [1]: from sympy.stats import *

In [2]: nu = Symbol("nu", positive=True)

In [3]: X = StudentT("x", nu)

In [4]: density(X)(x)
Out[4]: 
          ν   1         
        - ─ - ─         
          2   2         
⎛     2⎞                
⎜    x ⎟         ⎛ν   1⎞
⎜1 + ──⎟       ⋅Γ⎜─ + ─⎟
⎝    ν ⎠         ⎝2   2⎠
────────────────────────
      ___   ___  ⎛ν⎞    
    ╲╱ π ⋅╲╱ ν ⋅Γ⎜─⎟    
                 ⎝2⎠    

In [5]: t = Symbol('t', positive=True)

In [6]: simplify(E(exp(I*t*X)))
Out[6]: 
 ν                    ν ⎛      ⅈ⋅π⋅ν                                                                                       ⎞  -ⅈ⋅π⋅ν
 ─ - 1   ⎛4 ___   ___⎞  ⎜      ─────                                                                                       ⎟  ──────
 2       ⎜╲╱ ν ⋅╲╱ t ⎟  ⎜   ν    2      ⎛π⋅ν⎞        ⎛-ν    ___  ⎞    ν  ⅈ⋅π⋅ν        ⎛ν    ___  ⎞    ν        ⎛ν    ___  ⎞⎟    2   
2     ⋅π⋅⎜───────────⎟ ⋅⎜2⋅4 ⋅ℯ     ⋅cos⎜───⎟⋅besseli⎜──, ╲╱ ν ⋅t⎟ - 4 ⋅ℯ     ⋅besseli⎜─, ╲╱ ν ⋅t⎟ - 4 ⋅besseli⎜─, ╲╱ ν ⋅t⎟⎟⋅ℯ      
         ⎝     8     ⎠  ⎝               ⎝ 2 ⎠        ⎝2          ⎠                    ⎝2         ⎠             ⎝2         ⎠⎠        
────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
                                                          ⎛π⋅ν⎞    ⎛π⋅ν⎞  ⎛ν⎞                                                       
                                                       sin⎜───⎟⋅cos⎜───⎟⋅Γ⎜─⎟                                                       
                                                          ⎝ 2 ⎠    ⎝ 2 ⎠  ⎝2⎠