(f, x, t)
Compute the inverse laplace transform
    :math:\int_{c+i\infty}^{c-i\infty} f(x) e^{tx) dx,
    for real c larger than the real part of all singularities of f.
    Note that ``t`` is always assumed real and positive.

    Return None if the integral does not exist or could not be evaluated.

    >>> from sympy.abc import x, t
    >>> from sympy.integrals.meijerint import meijerint_inversion
    >>> meijerint_inversion(1/x, x, t)
    Heaviside(t)
Compute the inverse laplace transform
    :math:\int_{c+i\infty}^{c-i\infty} f(x) e^{tx) dx,
    for real c larger than the real part of all singularities of f.
    Note that ``t`` is always assumed real and positive.
Compute the inverse laplace transform : math : \ int_ { c + i \ infty } ^ { c - i \ infty } f ( x ) e^ { tx ) dx for real c larger than the real part of all singularities of f . Note that t is always assumed real and positive .
def meijerint_inversion(f, x, t):
    """
    Compute the inverse laplace transform
    :math:\int_{c+i\infty}^{c-i\infty} f(x) e^{tx) dx,
    for real c larger than the real part of all singularities of f.
    Note that ``t`` is always assumed real and positive.

    Return None if the integral does not exist or could not be evaluated.

    >>> from sympy.abc import x, t
    >>> from sympy.integrals.meijerint import meijerint_inversion
    >>> meijerint_inversion(1/x, x, t)
    Heaviside(t)
    """
    from sympy import I, Integral, exp, expand, log, Add, Mul, Heaviside
    f_ = f
    t_ = t
    t = Dummy('t', polar=True)  # We don't want sqrt(t**2) = abs(t) etc
    f = f.subs(t_, t)
    c = Dummy('c')
    _debug('Laplace-inverting', f)
    if not _is_analytic(f, x):
        _debug('But expression is not analytic.')
        return None
    # We filter out exponentials here. If we are given an Add this will not
    # work, but the calling code will take care of that.
    shift = 0
    if f.is_Mul:
        args = list(f.args)
        newargs = []
        exponentials = []
        while args:
            arg = args.pop()
            if isinstance(arg, exp):
                arg2 = expand(arg)
                if arg2.is_Mul:
                    args += arg2.args
                    continue
                try:
                    a, b = _get_coeff_exp(arg.args[0], x)
                except _CoeffExpValueError:
                    b = 0
                if b == 1:
                    exponentials.append(a)
                else:
                    newargs.append(arg)
            elif arg.is_Pow:
                arg2 = expand(arg)
                if arg2.is_Mul:
                    args += arg2.args
                    continue
                if x not in arg.base.free_symbols:
                    try:
                        a, b = _get_coeff_exp(arg.exp, x)
                    except _CoeffExpValueError:
                        b = 0
                    if b == 1:
                        exponentials.append(a*log(arg.base))
                newargs.append(arg)
            else:
                newargs.append(arg)
        shift = Add(*exponentials)
        f = Mul(*newargs)

    gs = _rewrite1(f, x)
    if gs is not None:
        fac, po, g, cond = gs
        _debug('Could rewrite as single G function:', fac, po, g)
        res = S(0)
        for C, s, f in g:
            C, f = _rewrite_inversion(fac*C, po*x**s, f, x)
            res += C*_int_inversion(f, x, t)
            cond = And(cond, _check_antecedents_inversion(f, x))
        cond = _my_unpolarify(cond)
        if cond == False:
            _debug('But cond is always False.')
        else:
            _debug('Result before branch substitution:', res)
            res = _my_unpolarify(hyperexpand(res))
            if not res.has(Heaviside):
                res *= Heaviside(t)
            res = res.subs(t, t + shift)
            if not isinstance(cond, bool):
                cond = cond.subs(t, t + shift)
            return Piecewise((res.subs(t, t_), cond),
                             (Integral(f_*exp(x*t), (x, c - oo*I, c + oo*I)).subs(t, t_), True))
def meijerint_inversion ( f , x , t ) : from sympy import I , Integral , exp , expand , log , Add , Mul , Heaviside f_ = f t_ = t t = Dummy ( 't' , polar = True ) # We don't want sqrt(t**2) = abs(t) etc f = f . subs ( t_ , t ) c = Dummy ( 'c' ) _debug ( 'Laplace-inverting' , f ) if not _is_analytic ( f , x ) : _debug ( 'But expression is not analytic.' ) return None # We filter out exponentials here. If we are given an Add this will not # work, but the calling code will take care of that. shift = 0 if f . is_Mul : args = list ( f . args ) newargs = [ ] exponentials = [ ] while args : arg = args . pop ( ) if isinstance ( arg , exp ) : arg2 = expand ( arg ) if arg2 . is_Mul : args += arg2 . args continue try : a , b = _get_coeff_exp ( arg . args [ 0 ] , x ) except _CoeffExpValueError : b = 0 if b == 1 : exponentials . append ( a ) else : newargs . append ( arg ) elif arg . is_Pow : arg2 = expand ( arg ) if arg2 . is_Mul : args += arg2 . args continue if x not in arg . base . free_symbols : try : a , b = _get_coeff_exp ( arg . exp , x ) except _CoeffExpValueError : b = 0 if b == 1 : exponentials . append ( a * log ( arg . base ) ) newargs . append ( arg ) else : newargs . append ( arg ) shift = Add ( * exponentials ) f = Mul ( * newargs ) gs = _rewrite1 ( f , x ) if gs is not None : fac , po , g , cond = gs _debug ( 'Could rewrite as single G function:' , fac , po , g ) res = S ( 0 ) for C , s , f in g : C , f = _rewrite_inversion ( fac * C , po * x ** s , f , x ) res += C * _int_inversion ( f , x , t ) cond = And ( cond , _check_antecedents_inversion ( f , x ) ) cond = _my_unpolarify ( cond ) if cond == False : _debug ( 'But cond is always False.' ) else : _debug ( 'Result before branch substitution:' , res ) res = _my_unpolarify ( hyperexpand ( res ) ) if not res . has ( Heaviside ) : res *= Heaviside ( t ) res = res . subs ( t , t + shift ) if not isinstance ( cond , bool ) : cond = cond . subs ( t , t + shift ) return Piecewise ( ( res . subs ( t , t_ ) , cond ) , ( Integral ( f_ * exp ( x * t ) , ( x , c - oo * I , c + oo * I ) ) . subs ( t , t_ ) , True ) )