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# addition formula

The addition formula of a real or complex function shows how the value of the function on a sum-formed variable can be expressed with the values of this function and perhaps of another function on the addends.

Examples

1. Addition formula of an additive function $f$,

$f(x\!+\!y)=f(x)+f(y)$2. Addition formula of the natural power function, i.e. the binomial theorem,

$(x\!+\!y)^{n}=\sum_{{j=0}}^{n}{n\choose j}x^{{n-j}}y^{j}\qquad(n=0,\,1,\,2,\,\ldots)$3. Addition formula of the exponential function,

$e^{{x+y}}=e^{x}e^{y}$4. Addition formulae of the trigonometric functions, e.g.

$\cos(x\!+\!y)=\cos{x}\cos{y}-\sin{x}\sin{y},\footnote{The addition formula of % cosine is sometimes called ``the mother of all formulae''.}\,\,\,\,\tan(x\!+\!% y)=\frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}$5. Addition formulae of the hyperbolic functions, e.g.

$\sinh(x\!+\!y)=\sinh{x}\cosh{y}+\cosh{x}\sinh{y}$6. Addition formula of the Bessel function,

$J_{n}(x\!+\!y)=\sum_{{\nu=-\infty}}^{{\infty}}J_{\nu}(x)J_{{n-\nu}}(y)\qquad(n% =0,\,\pm 1,\,\pm 2,\,\ldots)$

The five first of those are instances of algebraic addition formulae; e.g. $\cosh{x}$ and $\sinh{x}$ are tied together by the algebraic connection $\cosh^{2}{x}-\sinh^{2}{x}=1$.

One may also speak of the subtraction formulae of functions — one example would be $e^{{x-y}}=\frac{e^{x}}{e^{y}}$.

## Mathematics Subject Classification

30D05*no label found*30A99

*no label found*26A99

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## Comments

## why addition formula is attached to persistence?

May be I missed something, but what is relation of this entry to the entry "persistence of analytic relations"?

## Re: why addition formula is attached to persistence?

The attachment isn't so good.

## Re: Re: why addition formula is attached to persistence?

Well, the new attachment is also not that good to my opinion. I wouldn't attach it to anything.

## Re: Re: why addition formula is attached to persistence?

OK, if you are thought the thing thoroughly, so I can remove the attachement. I have not found any natural attachement. BTW, if you know some good additional exemples, please tell me!

Jussi

## Re: Re: why addition formula is attached to persistence?

Some other good examples would be the addition formulae for elliptic functions. In fact, Weierstrass showed that the only complex analytic functions which have an addition theorem are algebraic functions elliptic functions (or limiting cases such as trigonometric functions).

## Re: Re: why addition formula is attached to persistence?

> In fact, Weierstrass showed that

> the only complex analytic functions

> which have an addition theorem are

> algebraic functions, elliptic functions, or limiting cases

> (such as trigonometric functions).

So, then this is the point! I didn't know this theorem, but now it is clear, that this entry is naturally attached to this theorem. I guess, such theorem is not presented yet in encyclopedia, so I think it would be reasonable if not to make an entry for this theorem, then at least to mention it here (in the entry "addition formula").

## homogeneous & addition formula

Why is there a reference to homogeneous functions

in entry one:

L(x+y)=L(x)+L(y)

Doen't that addition formula hold for any linear function?

Matte

## Re: homogeneous & addition formula

The "linear function" means often such that the values are determined by a first degree polynomial (cf. e.g. http://en.wikipedia.org/wiki/Linear_function) when it is question of real functions. Therefore it is more certain to say "homogeneous linear".

This is in accordance with the entry "homogeneous function" --

such functions in general have a certain degree (the entry speaks of homogeneous functions of degree...); I wanted to speak of homog. function of degree 1, i.e. linear.

So I said "homogeneous linear function" in the entry "addition formula".

## Re: homogeneous & addition formula

Hi

Sometimes yes. However, I think that its use is

more common in areas like mathematical modelling

(a linear model, linear approximation, etc.)

The proper mathematical name for a mapping of the form

L(v)+v is an _affine transformation_. I

added an entry on this.

http://planetmath.org/?op=getobj&from=objects&name=AffineTransformation

On PM a _linear transformation_ is defined here:

http://planetmath.org/?op=getobj&from=objects&name=LinearTransformation

Thus, as a correction for this entry, "linear" should link

to some entry. Also, if it links to the above definition of

linear transformation, there is no need for reference to

the homogeneous entry.

Matte