Editing Eigenfunction (section)

===Derivative example===
A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space '''C'''<sup>∞</sup> of infinitely differentiable real or complex functions of a real or complex argument ''t''. For example, consider the derivative operator <math display="inline" alt="d over dt">\frac{d}{dt}</math> with eigenvalue equation
<math display="block" alt="the derivative of f of t equals lambda times f of t"> \frac{d}{dt}f(t) = \lambda f(t).</math>

This differential equation can be solved by multiplying both sides by <math display="inline" alt="dt over f of t">\frac{dt}{f(t)}</math> and integrating. Its solution, the [[exponential function]]
<math display="block" alt="f of t equals f nought times e raised to lambda t"> f(t)=f_0 e^{\lambda t},</math>
is the eigenfunction of the derivative operator, where ''f''<sub>0</sub> is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value. In particular, note that for λ = 0 the eigenfunction ''f''(''t'') is a constant.

Suppose in the example that ''f''(''t'') is subject to the boundary conditions ''f''(0) = 1 and <math display="inline" alt="df over dt at t equals 0 is 2">\left.\frac{df}{dt}\right|_{t=0} = 2</math>. We then find that
<math display="block" alt="f of t equals e raised to 2t"> f(t)=e^{2t},</math>
where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.