Take your wrapping function from t1 to t2, at a frequency ƒ (signal) != Fs (sampling freq), then take the limit as t1/t2 goes to -∞/+∞. As your window gets longer, more cycles of the oscillation "cancel out", moving the center of mass towards 0+0i. This means the peak around ƒ narrows and raises. At ∞, ƒ becomes infinitely narrow and high (Dirac delta * ).
This is also why peaks on an fft are gaussian (finite window), and get sharper as the fft window is increased.
* for cosine, technically there is a peak at -ƒ too. this is because a real cosine signal is ambiguous whether it is "moving forward or backwards in time". Hence it has a peak at +/-ƒ. A complex exponetial (helix through time) has chirality due to the real and imag components, so it has a single peak at ƒ. And if you take a +ƒ (lefthanded) and -ƒ helix (righthanded) and add them, the complex part cancels out, leaving only a real "up and down" wave.
This is also why peaks on an fft are gaussian (finite window), and get sharper as the fft window is increased.
* for cosine, technically there is a peak at -ƒ too. this is because a real cosine signal is ambiguous whether it is "moving forward or backwards in time". Hence it has a peak at +/-ƒ. A complex exponetial (helix through time) has chirality due to the real and imag components, so it has a single peak at ƒ. And if you take a +ƒ (lefthanded) and -ƒ helix (righthanded) and add them, the complex part cancels out, leaving only a real "up and down" wave.