I ran throughout the Clausen perform the opposite day, and after I noticed a plot of the perform my first thought was that it seems to be sorta like a sawtooth wave.
I puzzled whether or not it additionally seems like a sawtooth wave, and certainly it does. Extra on that shortly.
The Clausen perform might be outlined by way of its Fourier sequence:
The perform generally often known as the Clausen perform is considered one of a household of capabilities, therefore the subscript 2. The Clausen capabilities for all non-negative integers n are outlined by changing 2 with n on either side of the defining equation.
The Fourier coefficients decay quadratically, as do these of a triangle wave or sawtooth wave, as mentioned here. This means the perform Cl2(x) can not have a steady spinoff. Actually, the spinoff of Cl2(x) is infinite at 0. This follows shortly from the integral illustration of the perform.
The fundamental theorem of calculus exhibits that the spinoff
blows up at 0.
Now suppose we create an audio clip of Cl2(440x). This creates a sound with pitch A 440, however somewhat than a sinewave it has an disagreeable buzzing sound, very similar to a sawtooth wave.
The harshness of the sound is because of the gradual decay of the Fourier coefficients; the Fourier coefficients of extra nice musical sounds decay a lot quicker than quadratically.
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