Fourier transform of the convolution

I am trying to prove that an exponential input to a linear system is also exponential with the same frequency but different amplitude and phase.

That is, assuming that y(t)=L[x(t)]y(t)=L[x(t)], y(t)y(t) can be computed using the convolution functiony(t)=x(t)∗h(t)=∫∞−∞x(τ)h(t−τ)dτy(t)=x(t)*h(t)=\int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau, where x(t)x(t) is the input and h(t)h(t) is the system impulse response.

Now, let assume that x(t)=ejw0tx(t)=e^{jw_0t}. Then, according to linear system theory the corresponding output should be y(t)=H(w0)ejw0ty(t)=H(w_0)e^{jw_0t}, where H(w)H(w) is the fourier transform of h(t)h(t). Also it is known that x(t)∗h(t)=h(t)∗x(t)x(t)*h(t)=h(t)*x(t), that is convolution function is commutative.

I am doing the following steps in the proof:

y(t)=∫∞−∞x(τ)h(t−τ)dτ=∫∞−∞h(τ)x(t−τ)dτy(t)=\int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau=\int_{-\infty}^{\infty} h(\tau)x(t-\tau)d\tau
y(t)=∫∞−∞h(τ)ejw0(t−τ)dτy(t)=\int_{-\infty}^{\infty} h(\tau)e^{jw_0(t-\tau)}d\tau
since tt is independent of τ\tau
y(t)=ejw0t∫∞−∞h(τ)e−jw0τdτ⏟H(w0)−>y(t)=H(w0)ejw0ty(t)=e^{jw_0t}\underbrace{\int_{-\infty}^{\infty} h(\tau)e^{-jw_0\tau}d\tau}_{H(w_0)}->y(t)=H(w_0)e^{jw_0t}

Up to now everything is fine. However, the problem starts when I change the limits of the convolution integral. Assuming that linear system is causal, i.e. x(t)=0,h(t)=0x(t)=0, h(t)=0 for t<0t<0, the convolution integral becomes: y(t)=∫t0x(τ)h(t−τ)dτ=∫t0h(τ)x(t−τ)dτy(t)=\int_{0}^{t} x(\tau)h(t-\tau)d\tau=\int_{0}^{t} h(\tau)x(t-\tau)d\tau By applying the same logic above, y(t)=∫t0h(τ)ejw0(t−τ)dτy(t)=\int_{0}^{t} h(\tau)e^{jw_0(t-\tau)}d\tau since tt is independent of τ\tau y(t)=ejw0t∫t0h(τ)e−jw0τdτ⏟≠H(w0)y(t)=e^{jw_0t}\underbrace{\int_{0}^{t} h(\tau)e^{-jw_0\tau}d\tau}_{\ne H(w_0)} The integral does not equal to H(w0)H(w_0) since the integration depends on tt. I believe that the first derivation is correct but could not figure out what is wrong in my second derivation. Can you please help me to figure out the error I am making in the second derivation? Thanks in advance. Regards, ================= ================= =================