Weierstrasse's decomposition formula

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The Weierstrass apportionment formula is a formula from the real Analysis and goes back to the German mathematician Karl Weierstrass . It breaks down values ​​of differentiable functions into two summands: firstly, the value of the tangent function of the output function with respect to a point of the domain and secondly, the remainder or the error of the linear approximation .

This formula is fundamental in differential calculus, since Weierstrasse's decomposability is equivalent to the fundamental property of the differentiability of functions. It is exemplary of Weierstrass' services to the systematization and exactification of analysis.

Meaning of the formula

The representation of the function values ​​by the Weierstrasse decomposition formula is usually done with the help of a function of two variables, which is defined accordingly . Their function values ​​indicate the difference in value between the tangent function and the function , the graph being touched by the in : ${\displaystyle R_{f}}$ ${\displaystyle f}$ ${\displaystyle t}$ ${\displaystyle f}$ ${\displaystyle t}$ ${\displaystyle f}$ ${\displaystyle x_{o}}$

${\displaystyle f(x)=t(x,x_{o})+R(x,x_{o})=f'(x_{o})\cdot (x-x_{o})+f(x_{o})+R_{f}(x,x_{o})}$

The importance of this formula lies in the nature of the residual limb : The exact course of the function is indeed often uninteresting, however, is significant that they are in an environment of defined and the border crossing with higher than linear order to converge (cf.. Convergence speed ). Therefore it can be rewritten as follows: with . There are some useful aspects: ${\displaystyle R_{f}(x,x_{o})}$ ${\displaystyle R_{f}}$ ${\displaystyle x_{o}}$ ${\displaystyle x\to x_{o}}$ ${\displaystyle 0}$ ${\displaystyle R_{f}}$ ${\displaystyle R_{f}(x,x_{o})=r_{f}(x,x_{o})\cdot (x-x_{o})}$ ${\displaystyle \textstyle \lim _{x\to x_{o}}r_{f}(x,x_{o})=0}$

Approximation quality

Because of the square convergence of the residual limb, the tangent function is itself the optimal local linear approximation of the function with respect to . The attribute “local” expresses that in general exactly the arguments from a (depending on the competing approximation function) sufficiently small neighborhood of provide the better approximation of the function values. This behavior is used, for example, with the known approximation formulas and for arguments in a small environment of . ${\displaystyle t}$ ${\displaystyle f}$ ${\displaystyle x_{o}}$ ${\displaystyle x_{o}}$ ${\displaystyle \sin x\approx x}$ ${\displaystyle \tan x\approx x}$ ${\displaystyle 0}$

Alternative definition of differentiability

The equivalence of differentiability and Weierstrassian decomposability enables a different notation for the property of differentiability and thus a different approach to infinitesimal calculus as an alternative to the existence statement about the differential quotient . This notation enables, for example, a brief proof of the chain rule , which is elementary in analysis , while the difference quotient notation has its pitfalls here .

proof

The equivalence of Weierstraß's decomposability and differentiability is shown by the proof of the implication in both directions.

Conclusion from differentiability to decomposability

It is shown that when a differentiable function is decomposed in the manner mentioned, the remainder of the term actually converges faster than linearly and can therefore be represented in the notation using the function . ${\displaystyle \ 0}$ ${\displaystyle \ r_{f}}$

Let with be chosen from a neighborhood of in which is defined and be differentiable in . Then ${\displaystyle \ x}$ ${\displaystyle x\neq x_{o}}$ ${\displaystyle \ x_{o}}$ ${\displaystyle \ f}$ ${\displaystyle \ f}$ ${\displaystyle \ x_{o}}$

${\displaystyle \ f(x)=f'(x_{o})\cdot (x-x_{o})+f(x_{o})+R_{f}(x,x_{o})}$

${\displaystyle \Longleftrightarrow {\frac {f(x)-f(x_{o})-f'(x_{o})\cdot (x-x_{o})}{x-x_{o}}}={\frac {R_{f}(x,x_{o})}{x-x_{o}}}.}$

Since in is differentiable, the left side of the equation for converges and the desired property of the remainder is obtained: ${\displaystyle \ f}$ ${\displaystyle \ x_{o}}$ ${\displaystyle x\to x_{o}}$

${\displaystyle \lim _{x\to x_{o}}\left({\frac {f(x)-f(x_{o})}{x-x_{o}}}-f'(x_{o})\right)=0=\lim _{x\to x_{o}}{\frac {R_{f}(x,x_{o})}{x-x_{o}}}}$

This is the proof for . The function values can therefore be decomposed in the following way: ${\displaystyle x\neq x_{0}}$ ${\displaystyle \ f(x)}$

${\displaystyle \ f(x)-f(x_{o})=(f'(x_{o})+r_{f}(x,x_{o}))\cdot (x-x_{o}),\lim _{x\to x_{o}}r_{f}(x,x_{o})=0}$

Note: The value can be defined. The decomposition formula is therefore valid for trivially and respectively are defined with respect to the first variable in continuous and, as mentioned above, in a neighborhood of . ${\displaystyle \ r_{f}(x_{o},x_{o}):=0}$ ${\displaystyle \ x=x_{o}}$ ${\displaystyle \ R_{f}}$ ${\displaystyle \ r_{f}}$ ${\displaystyle \ x_{o}}$ ${\displaystyle \ x_{o}}$

Conclusion from separability to differentiability

The starting point is the decomposition formula for the function , where the term whose existence is the assertion is replaced by a real value of a suitably defined function . Be chosen as in the previous proof and be a function dependent on . ${\displaystyle \ f}$ ${\displaystyle \ f'(x_{o})}$ ${\displaystyle \ c(x_{o})}$ ${\displaystyle \ c}$ ${\displaystyle \ x}$ ${\displaystyle \ r_{f}}$ ${\displaystyle \ f}$ ${\displaystyle \lim _{x\to x_{o}}r_{f}(x,x_{o})=0}$

So: ${\displaystyle f(x)=c(x_{o})\cdot (x-x_{o})+f(x_{o})+r_{f}(x,x_{0})\cdot (x-x_{o})}$

${\displaystyle \Longleftrightarrow {\frac {f(x)-f(x_{o})}{x-x_{o}}}=c(x_{o})+r_{f}(x,x_{o}).}$

The right-hand side of the equation converges for the limit crossing , since it only depends on and thus the differential quotient exists. ${\displaystyle \ x\to x_{o}}$ ${\displaystyle \ c}$ ${\displaystyle \ x_{o}}$

In addition, there is even what enables a determination of the derivative values ​​of a function as an alternative to the difference quotient. ${\displaystyle \ c(x_{o})=f'(x_{o})}$

Other spellings

• Analogous to the variants of the notation of the difference quotient, the above notation of the decomposition formula with the reference point and the variable can also be expressed using and the difference or between the variable and the reference point. Then the decomposition formula is called Here, with regard to the values ​​of , it should be noted that the property of at least quadratic vanishing occurs for the boundary transition of the arguments , i.e. is consequently considered in an area of with regard to the first variable. ${\displaystyle x_{o}}$ ${\displaystyle x}$ ${\displaystyle x_{o}}$ ${\displaystyle h}$ ${\displaystyle \Delta x}$ ${\displaystyle \ f(x_{o}+h)=f'(x_{o})\cdot h+f(x_{o})+R_{f}(h,x_{o}).}$ ${\displaystyle R_{f}}$ ${\displaystyle h\to 0}$ ${\displaystyle (h,x_{o})}$ ${\displaystyle R_{f}}$ ${\displaystyle 0}$

• In addition, one can also define as a function of only one variable, namely in the sense used above (or with the varied notation), if one always points out when using it that one is only using a special and constant one . ${\displaystyle R_{f}}$ ${\displaystyle x}$ ${\displaystyle h}$ ${\displaystyle x_{o}}$

• A notation with suggests that the formula should also be formulated: In this case, however, the misunderstanding of as the actual function value difference of can occur, although in reality only the linear increase is meant. This problem can be avoided by using Leibniz's differential notation with and instead of the delta notation , which basically results from the local decomposition of the function. ${\displaystyle \Delta x}$ ${\displaystyle \Delta y:=f(x_{o}+\Delta x)-f(x_{0})}$ ${\displaystyle \Delta y=f'(x_{o})\cdot \Delta x+R_{f}(\Delta x,x_{0}).}$ ${\displaystyle \Delta y}$ ${\displaystyle f}$ ${\displaystyle dx}$ ${\displaystyle dy}$

See also

• Taylor formula