Periodic smooth alternating series other than sin and cos

[Matt Benesi]

1) Are there any periodic alternating series functions other than sine and cosine (and series derived from them, like the series for cos(a) * cos(b))?

2) What is the following series called when x is (0,1) and (1,2]? Quasiperiodic? Semi?
$$\sum_{n=0}^\infty \, (-1)^n \, \frac{a^{x+2n}}{\Gamma (x+2n+1)}$$

3) What is the formula for the fluctuating "period" for the above series?

4) are there any (quasi/semi) periodic alternating series that do not use the gamma function for successive terms t_n, with ratio a, x=0 or 1?

$$\sum_{n=0}^\infty \, (-1)^n \, \frac{a^{n+x}}{t_n}$$

 

[mfb]

1. You can write every smooth periodic function as combination of sine and cosine functions. That is the idea of a Fourier transformation.

I don't understand the other questions.

 

[BvU]

Bessel functions, Associated Legendre polynomials, Hermite functions, ...

 

[Matt Benesi]

1. You can write every smooth periodic function as combination of sine and cosine functions.

Didn't want to do that:
"no series derived from them (sine and cosine), like the series for cos(a) * cos(b)".

thank you though.

 

[mfb]

You asked if there are functions with a specific property, I explained why there are no functions with this property.

 

[Matt Benesi]

Bessel functions, Associated Legendre polynomials, Hermite functions, ...

Bessel looks like the gamma function thing I posted. It doesn't look like it is periodic with x>2 (for the version I posted).

Legendre Polynomials? I looked them up and they don't look like they are periodic?

Hermite functions don't look periodic either.

 

[Matt Benesi]

You asked if there are functions with a specific property, I explained why there are no functions with this property.

So your answer to 1 is no.

For 2, I wanted to know what to call the ?Bessel? function I posted under 2. Quasiperiodic? Semiperiodic? Sort of periodic? Not really periodic? Almost periodic?

For 3, the period between zeroes varies. Maybe the period for maxes and minimums does not. I was asking for a specific... I have to leave. Sorry. Back later.

 

[Klystron]

An interesting function that meets the 'spirit' of your criteria: e^(-x^2)). The Taylor series inverts sign at each term:

Taylor series expansion http://www.efunda.com/math/taylor_series/exponential.cfm

And a video with derivations

No trig functions used.

 

[suremarc]

I believe the series in (2) is just the order-x fractional integral of cosine, which turns out just to be cos(t-xπ/2). Hence, by the angle addition formula it has a very simple expression as a sum of a sine wave and a cosine wave.

 

[BvU]

don't look like they are periodic?
...
Hermite functions don't look periodic either.

Missed the strictness of the 'periodic' constraint . Second @mfb #2

 

[Matt Benesi]

1) Are there any periodic alternating series functions other than sine and cosine (and series derived from them, like the series for cos(a) * cos(b))?

1) mfb answered that all smooth periodic functions (and thus alternating series) can be written as a combination of sine and cosine functions (alternating series):

This means each term of cosine or sine would just be something like (for cosine only series, assume +pi/2 or something for sine)):
$$x_1 \, cos(a_1) + x_2 \, cos(a_2) +...+x_m \, cos(a_m) =\\ \, \\ 1 - \frac {x_1 \, a_1^2 + x_2 \,a_2^2+..+ x_m \,a_m^2}{2!} + \frac {x_1 \, a_1^4 + x_2 \,a_2^4+..+ x_m \,a_m^4}{4!}-... =\\ \\ \, \\ 1 +\sum_{t=1}^{\infty} (-1)^t \,\, \frac{\sum_{k=1}^{m} x_m \, a_m^{2t}}{(2t)!}$$

ETA: So, this uses the gamma function $\Gamma \, (2t+1)$, so to rephrase my question, is there a way to rewrite the above that uses something other than the gamma function for successive terms?

Something like g(t) replaces gamma function, $a^{f(t)}$ or simply f(a,t) replaces a^2t?

$$\sum_{t=1}^\infty \, (-1)^{t+1} \, \frac{f(a,t)}{g(t)}$$

2) What is the following series called when x is (0,1) and (1,2]? Quasiperiodic? Semi?
$$\sum_{n=0}^\infty \, (-1)^n \, \frac{a^{x+2n}}{\Gamma (x+2n+1)}$$

So this looks a bit like a Bessel (except the 2n), according to suremarc the partial integral of cosine/sine, but I still do not know what to call the results of the series when x!=0 or 1 (x is not 0 or 1).

Would it be called quasiperiodic (because the periods aren't exactly the same?)? Semiperiodic? Metaperiodic? Almost periodic? What is the proper word for functions that are almost periodic, such as:

$$cos (\frac{|cos(t)+cos(t\,\times\, cos(t))|}{20}+t)$$

3) What is the formula for the fluctuating "period" for the above series?

Just curious about the function in question 2, if someone knows the answer or feels like working out their brain a bit.

4) are there any (quasi/semi) periodic alternating series that do not use the gamma function (or factorials) for successive terms t_n, with ratio a,[strike] x=0 (sine) or 1 (cosine)[/strike]?

$$\sum_{n=0}^\infty \, (-1)^n \, \frac{a^{n+x}}{t_n}$$

This was basically answered by mfb in their response to 1.

If every smooth periodic function can be represented by a finite amount of cosine and sine functions the answer is No.

However, if some cases require an infinite amount of sine/cosine functions, would the answer to both 1 and 4 be maybe? That would make sine/cosine superfluous, right? I'm just looking for alternating series that don't use gamma/factorials between successive terms that are semi periodic (not increasing or decreasing a lot, big O of 1 is fine).

Thanks, Matt

 

[suremarc]

So I have to correct myself—the fractional integral of cosine or sine turns out to a complicated mess involving the hypergeometric function. However the Riemann-Liouville integral is a Fourier multiplier, so the series is still periodic, I believe.

If every smooth periodic function can be represented by a finite amount of cosine and sine functions

Be careful. He never said that it had to be a finite number. Most interesting functions have Fourier series which have infinitely many terms (for example, square waves).

However, if some cases require an infinite amount of sine/cosine functions, would the answer to both 1 and 4 be maybe?

It mostly depends on how you think about it. But it’s important to know that every periodic function has a (potentially infinite) Fourier series. So asking what kinds of periodic functions there are is equivalent to asking what kinds of sequences of Fourier coefficients are there.

 

[mfb]

is there a way to rewrite the above that uses something other than the gamma function for successive terms?

You found one way already, and you can always add, multiply, ... stuff to make it look different.

Here is a smooth periodic function in $a$ where you don't see any gamma function appearing: $\displaystyle f(a) = \sum_{n=0}^\infty \, (-1)^n \, \frac{\cos(an)}{n^n}$

You can still rewrite it to have factorials in if you want, of course: $\displaystyle f(a) = \sum_{n=0}^\infty \, (-1)^n \, \frac{\cos(an)n!/n^n}{n!}$

 

[Buzz Bloom]

Hi mfb:

You have a typo in the bottom equation. nⁿ should be in the denominator.

I apologize. I missed the "/".

Regards,
Buzz

 

[Matt Benesi]

You found one way already, and you can always add, multiply, ... stuff to make it look different.

Here is a smooth periodic function in $a$ where you don't see any gamma function appearing: $\displaystyle f(a) = \sum_{n=0}^\infty \, (-1)^n \, \frac{\cos(an)}{n^n}$

I see a gamma in the above, unless cos is an abbreviation I am not familiar with that looks like cosine? Or is cos a special form of cosine that doesn't look like: $\sum\limits_{t=0}^\infty \frac{a^{2t}} {\Gamma {(2t+1)} }$

You can still rewrite it to have factorials in if you want, of course: $\displaystyle f(a) = \sum_{n=0}^\infty \, (-1)^n \, \frac{\cos(an)n!/n^n}{n!}$

lol...

 

[mfb]

lol...

But that's the type of rewriting you want to do with the cosine, too...
Replace it by a triangle function if you want.

 

[Matt Benesi]

But that's the type of rewriting you want to do with the cosine, too...
Replace it by a triangle function if you want.

Ok, so every periodic function can be approximated with a certain amount of cosine, or sine, or combination thereof.

That's not the same thing as what's in 4, and I was wrong to say it basically was (and there was a typo in question 4, corrected here, should have been $a^{2n+x}$ not $a^{n+x}$, and I'd like to include a function $f(a)$ related to the magnitude of a):

4) are there any (quasi/semi) periodic alternating series that do not use the gamma function for successive terms t_n, with ratio a, x=0 or 1?

$$f(a) \,\, \times \,\, \sum_{n=0}^\infty \, (-1)^n \, \frac{a^{2n+x}}{t_n}$$

Trivially, any of these can be written with cosine/sine. Very awesome.

Are there series like what is written above that do not use the gamma function (so there is some function that generates $t_n$ that is not the gamma function).

I'm thinking there are an infinite amount O(a), however, I'm wondering about the special case O(1).