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addition formula (Definition)

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^xe^y$
  4. Addition formulae of the trigonometric functions, e.g.
    $ \cos(x+y) = \cos{x}\cos{y}-\sin{x}\sin{y},\,\,\,\, \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,\,\pm1,\,\pm2,\,\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}$.



"addition formula" is owned by pahio.
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See Also: example of solving a functional equation, proof of addition formula of exp, addition and subtraction formulas for sine and cosine, addition and subtraction formulas for tangent, addition and subtraction formulas for hyperbolic functions

Also defines:  addition formulae, subtraction formula, subtraction formulae
Keywords:  algebraic addition formula

Attachments:
Weierstrass' theorem on addition formulas (Theorem) by rspuzio
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Cross-references: algebraic, Bessel function, hyperbolic functions, binomial theorem, natural power function, additive function, variable, function, complex function
There are 17 references to this entry.

This is version 19 of addition formula, born on 2004-10-15, modified 2008-05-19.
Object id is 6374, canonical name is AdditionFormula.
Accessed 9281 times total.

Classification:
AMS MSC: 26A99 (Real functions :: Functions of one variable :: Miscellaneous)
 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
 30D05 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Functional equations in the complex domain, iteration and composition of analytic functions)
Pending Errata and Addenda
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Discussion
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homogeneous & addition formula by matte on 2004-10-24 04:10:50
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
[ reply | up ]
why addition formula is attached to persistence? by mathforever on 2004-10-15 16:01:36
May be I missed something, but what is relation of this entry to the entry "persistence of analytic relations"?
[ reply | up ]
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