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> It's sort of like, if you took one graph and flipped it over 180 degrees, and cut out the part where the two functions start converging or start diverging, an

> It's sort of like, if you took one graph and flipped it over 180 degrees, and cut out the part where the two functions start converging or start diverging, and ignore the units of measure, then the two graphs almost look similar... > So maybe they're related in some way... maybe there's something there... The two Airy functions, like the two Fresnel integrals, the Bessel functions, sine and cosine, and other pairs of popular functions, are the real and imaginary part of a very smooth complex function of one real variable, with the typical appearance of perfectly interleaved oscillations with the same approximate period and a delay of 1/4 period between the two functions. While sine and cosine represent a perfect specimen of this behaviour, other functions can have peculiarities: Fresnel integrals oscillate around the asymptotes rather than around 0, Airy functions cease to be approximately periodic around infinity, and so on.

peter_d_sherman 4y ago
Hi! Some excellent comments! I'm going to highlight them and add to them a little bit -- I hope that's OK with you [my modifications will be in brackets!]: >The two Airy functions, like the two Fresnel integrals, the Bessel functions, sine and cosine, and other pairs of popular functions, are the real and imaginary part of a very smooth complex function of one real variable , with the typical appearance of perfectly interleaved oscillations with the same approximate period and a delay of 1/4 period [well, some period that doesn't throw the two functions completely out of sync -- at least not immediately!] between the two functions. While sine and cosine represent a perfect specimen of this behaviour [Indeed! A very important and excellent point! (Also cosine is the perfect derivative of sine, negative sine is the perfect derivative of cosine, negative cosine is the perfect derivative of negative sine, and sine is the perfect derivative of negative cosine! In other words, derive or integrate one of those functions 4 times -- and you're back to the original function you started from!] , other functions can [and do!] have peculiarities [are not quite as mathematically perfect as sin/cos!]: Fresnel integrals oscillate around the asymptotes rather than around 0, Airy functions cease to be approximately periodic [cease to oscillate] around infinity, and so on." PDS: This leads me to wanting to research the following idea: Do sin and cos (and by extension, -sin and -cos) -- have a simple common function (or variable, complex or otherwise, as you alluded to) -- from which both (and by later extension, all) of them can be derived? Or are sin and cos (and by extension, -sin and -cos) sort of "mathematical absolutes" (can't be derived from anything except themselves?) for lack of a better term? ? Anyway, excellent post!
- HelloNurse 4y ago
  Sine and cosine are a bit special because they are the real and imaginary part of the exponential function of an imaginary argument; that's why they reappear when taking the derivative multiple times, why they are exactly periodic, and why the sum of their squares equals 1. Tightly wound spirals, like those formed by the Airy functions and the Fresnel integrals, have almost the same shape as exact repetitions of a circle; graphs end up looking very similar.

peter_d_sherman 4y ago

Hi! Some excellent comments! I'm going to highlight them and add to them a little bit -- I hope that's OK with you [my modifications will be in brackets!]: >The two Airy functions, like the two Fresnel integrals, the Bessel functions, sine and cosine, and other pairs of popular functions, are the real and imaginary part of a very smooth complex function of one real variable , with the typical appearance of perfectly interleaved oscillations with the same approximate period and a delay of 1/4 period [well, some period that doesn't throw the two functions completely out of sync -- at least not immediately!] between the two functions. While sine and cosine represent a perfect specimen of this behaviour [Indeed! A very important and excellent point! (Also cosine is the perfect derivative of sine, negative sine is the perfect derivative of cosine, negative cosine is the perfect derivative of negative sine, and sine is the perfect derivative of negative cosine! In other words, derive or integrate one of those functions 4 times -- and you're back to the original function you started from!] , other functions can [and do!] have peculiarities [are not quite as mathematically perfect as sin/cos!]: Fresnel integrals oscillate around the asymptotes rather than around 0, Airy functions cease to be approximately periodic [cease to oscillate] around infinity, and so on." PDS: This leads me to wanting to research the following idea: Do sin and cos (and by extension, -sin and -cos) -- have a simple common function (or variable, complex or otherwise, as you alluded to) -- from which both (and by later extension, all) of them can be derived? Or are sin and cos (and by extension, -sin and -cos) sort of "mathematical absolutes" (can't be derived from anything except themselves?) for lack of a better term? ? Anyway, excellent post!

HelloNurse 4y ago
Sine and cosine are a bit special because they are the real and imaginary part of the exponential function of an imaginary argument; that's why they reappear when taking the derivative multiple times, why they are exactly periodic, and why the sum of their squares equals 1. Tightly wound spirals, like those formed by the Airy functions and the Fresnel integrals, have almost the same shape as exact repetitions of a circle; graphs end up looking very similar.