Help please determine if the function is an eigenfunction of the operator. If so, determine the eigenvalue. function: e^(-i4x)

Question

determine if the function is an eigenfunction of the operator. If so, determine the eigenvalue.

function: e^(-i4x)

operator: -i  d/dx

Answer 1

Let f be the given function and F be the given operator.

F(f) = -i×df/dx = -i×(-i4)×e^-i4x = -4×e^-i4x = -4×f

Since the result is a constant multiple of f, the function is a eigenfunction of the operator.

And the constant -4 is the eigenvalue.