Analysis formula collection

${\displaystyle {\sqrt[{n}]{x}}}$

This article is a collection of formulas for analysis . Math symbols are used, which are explained in the article List of Math Symbols .

Sequences and ranks

Arithmetic and Geometric Sequences

Arithmetic sequence

${\displaystyle a_{n+1}-a_{n}=d\quad \mathrm {f{\ddot {u}}r\;alle\;} n}$

${\displaystyle a_{n}={\tfrac {1}{2}}(a_{n-1}+a_{n+1})}$

${\displaystyle \,a_{n}=a_{1}+(n-1)d}$

Geometric sequence

${\displaystyle {\frac {a_{n+1}}{a_{n}}}=q\quad \mathrm {f{\ddot {u}}r\;alle\;} n,q\in \mathbb {R} \setminus 0}$

${\displaystyle a_{n}={\sqrt {a_{n-1}\cdot a_{n+1}}}}$

${\displaystyle a_{n}=a_{1}\cdot q^{n-1}}$

Limit values : definition (consequences)

• The sequence is called zero sequence if there is a number for each , so that for all subsequent members with a higher number, the following applies: ${\displaystyle (a_{n})}$ ${\displaystyle \epsilon >0}$ ${\displaystyle n_{0}}$ ${\displaystyle n>n_{0}}$

${\displaystyle \,|a_{n}|<\epsilon }$

• A sequence has the limit value a if the sequence has the limit value 0. ${\displaystyle (a_{n})}$ ${\displaystyle (a_{n}-a)}$
• Sequences without a limit value are called divergent.
• A sequence is called restricted if there is a number such that applies to all . ${\displaystyle K>0}$ ${\displaystyle |f_{n}|<K}$ ${\displaystyle n\in \mathbb {N} }$

Limit sets (consequences)

If the sequence has the limit value a and the sequence has the limit value b, then: ${\displaystyle (a_{n})}$ ${\displaystyle (b_{n})}$

• ${\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=a\pm b}$
• ${\displaystyle \lim _{n\to \infty }(a_{n}\cdot b_{n})=a\cdot b}$
• ${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\frac {a}{b}}\qquad b\not =0}$

Functions (formal properties)

Limits of functions

• [Definition, properties, limit sets analogous]

Rule of de l'Hospital

Be ${\displaystyle f(x)={\frac {u(x)}{v(x)}}.}$

Requirements:

• There is a passage so and are either zero or determined diverge ${\displaystyle a}$ ${\displaystyle u(a)}$ ${\displaystyle v(a)}$
• ${\displaystyle u}$and are differentiable in a neighborhood of ${\displaystyle v}$ ${\displaystyle a}$
• The limit exists. ${\displaystyle \lim _{x\to a}{\frac {u'(x)}{v'(x)}}}$

Then:

${\displaystyle \lim _{x\to a}{\frac {u(x)}{v(x)}}=\lim _{x\to a}{\frac {u'(x)}{v'(x)}}}$

Unilateral limits

The function has for the limit when it (however small) to each one are such that for all values from the domain of satisfying the condition , also suffice applies. ${\displaystyle f\colon X\to \mathbb {R} \;}$ ${\displaystyle x\to p+}$ ${\displaystyle L\;}$ ${\displaystyle \varepsilon >0\;}$ ${\displaystyle \delta >0\;}$ ${\displaystyle x\;}$ ${\displaystyle X\;}$ ${\displaystyle f\;}$ ${\displaystyle 0<x-p<\delta \;}$ ${\displaystyle |f(x)-L|<\varepsilon \;}$

In this case the limit is called convergent . ${\displaystyle \lim _{x\to p+}f(x):=L}$

continuity

A function is called continuous at one point if the limit of for against exists and matches the function value ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x_{0})}$

${\displaystyle f(x_{0})=\lim _{h\to 0}f(x_{0}+h)=\lim _{h\to 0}f(x_{0}-h)=\lim _{x\to x_{0}}f(x)}$

• Epsilon-delta criterion : is continuous at if, at any one exists, such that for all of the following applies: . ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle x_{0}\in D}$
  ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle x\in D}$ ${\displaystyle |x-x_{0}|<\delta }$ ${\displaystyle |f(x)-f(x_{0})|<\varepsilon }$
• Sequence criterion : is continuous in if, for every sequence with elements that converges against , also converges against . ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle x_{0}\in D}$ ${\displaystyle (x_{k})_{k\in \mathbb {N} }}$ ${\displaystyle x_{k}\in D}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x_{k})}$ ${\displaystyle f(x_{0})}$

Basics

Intermediate value theorem

A function continuous in the interval ( ) takes every function value between and at least once. ${\displaystyle [a,b]}$ ${\displaystyle a<b}$ ${\displaystyle f}$ ${\displaystyle f(a)}$ ${\displaystyle f(b)}$

Special case: zero digit rate

A continuous function where and have different signs has at least one zero there. ${\displaystyle I}$ ${\displaystyle f(a)}$ ${\displaystyle f(b)}$

Extreme value theorem

A function that is continuous in an interval always has a largest and a smallest function value there.

Mean theorem

Let it be continuous and differentiable on the closed interval ( ). Then there is at least one such that ${\displaystyle f:[a,b]\to \mathbb {R} }$ ${\displaystyle [a,b]}$ ${\displaystyle a<b}$ ${\displaystyle x_{0}\in (a,b)}$

${\displaystyle f'\left(x_{0}\right)={\frac {f\left(b\right)-f\left(a\right)}{b-a}}}$

applies.

Differential calculus

Differentiability: Definitions

A function is differentiable at one point in its domain if the limit value ${\displaystyle f}$ ${\displaystyle x_{0}}$

${\displaystyle \lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}$

exists. This limit is called the derivative of at that point . ${\displaystyle f}$ ${\displaystyle x_{0}}$

Geometric: tangents

Tangent equation to in the point ${\displaystyle f}$ ${\displaystyle P(x_{0}|f(x_{0}))}$

${\displaystyle y=f'(x_{0})\!\,(x-x_{0})+f(x_{0})}$

Normal (vertical)

${\displaystyle y={\frac {-1}{f'(x_{0})}}(x-x_{0})+f(x_{0})}$

Derivation rules

Constant function

${\displaystyle \left(a\right)'=0}$

Factor rule

${\displaystyle (a\cdot f)'=a\cdot f'}$

Sum rule

${\displaystyle \left(g\pm h\right)'=g'\pm h'}$

Product rule

${\displaystyle (g\cdot h)'=g'\cdot h+g\cdot h'}$

Quotient rule

${\displaystyle \left({\frac {g}{h}}\right)'={\frac {g'\cdot h-g\cdot h'}{h^{2}}}}$

Power rule

${\displaystyle \left(x^{n}\right)'=nx^{n-1}}$

Chain rule

${\displaystyle (g\circ h)'(x)=(g(h(x)))'=g'(h(x))\cdot h'(x)}$

Derivation of the power function ${\displaystyle f(x)=g(x)^{h(x)}}$

${\displaystyle f'(x)=\left(h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}\right)g(x)^{h(x)}}$.

Leibniz rule

The derivation of -th order for a product of two -fold differentiable functions and results from ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(k)}g^{(n-k)}}$.

The expressions of the form appearing here are binomial coefficients . ${\displaystyle {\tbinom {n}{k}}}$

Formula by Faà di Bruno

This formula enables the closed representation of the -th derivative of the composition of two -fold differentiable functions. It generalizes the chain rule to higher derivatives. ${\displaystyle n}$ ${\displaystyle n}$

Deriving important functions

see table of derivative and antiderivative functions

Geometrical applications: properties of curves ( curve discussion )

Is looked at ${\displaystyle f\colon x\mapsto f(x)}$

Investigation aspect	criteria
Zero	${\displaystyle f(x_{N})=0\,}$
Extreme value	${\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})\neq 0}$
minimum	${\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})>0}$
maximum	${\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})<0}$
Turning point	${\displaystyle f''(x_{W})=0\quad {\text{und}}\quad f'''(x_{W})\neq 0}$
Saddle point	${\displaystyle f'(x_{W})=0\quad {\text{und}}\quad f''(x_{W})=0\quad {\text{und}}\quad f'''(x_{W})\neq 0}$
Behavior in infinity	${\displaystyle \lim _{x\to \infty }f(x)\quad {\text{und}}\quad \lim _{x\to -\infty }f(x)}$
symmetry
Axis symmetry to the coordinate axis ("straight")	${\displaystyle f(x)=f(-x)\,}$
Point symmetry to the origin of coordinates ("odd")	${\displaystyle -f(x)=f(-x)\,}$
monotony
monotonically increasing or strictly monotonically increasing	${\displaystyle f'(x)\geq 0\quad {\text{bzw.}}\quad f'(x)>0\,}$
falling monotonically or strictly falling monotonically	${\displaystyle f'(x)\leq 0\quad {\text{und}}\quad f'(x)<0}$
curvature
Left curve / convex arch (open at the top)	${\displaystyle f''(x)\geq 0\,}$
Right curve / concave arch (open at the bottom)	${\displaystyle f''(x)\leq 0\,}$
periodicity	${\displaystyle f(x+p)=f(x)\,}$

Fractional rational functions

Function term:

${\displaystyle f(x)={\frac {a_{z}x^{z}+a_{z-1}x^{z-1}+\cdots +a_{1}x+a_{0}}{b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots +b_{1}x+b_{0}}}={\frac {P_{z}(x)}{Q_{n}(x)}}}$

• Classification
  • If the denominator polynomial is of degree 0 (i.e. n = 0 and b ₀ ≠ 0) and is not the zero polynomial , one speaks of an entirely rational or a polynomial function . ${\displaystyle Q_{n}}$ ${\displaystyle P_{z}}$
  • If n > 0, it is a fractional-rational function .
  • If n > 0 and z < n , then it is a really fractional-rational function .
  • If n > 0 and z ≥ n , then it is an incorrect fractional rational function . It can be divided into a fully rational function and a truly fractional-rational function using polynomial division .
• Domain of definition
  • ${\displaystyle \mathbb {D} =\mathbb {R} \setminus \lbrace x_{0}\mid Q_{n}(x_{0})=0\rbrace }$
• Asymptotic behavior: For strives ${\displaystyle x\to \infty }$ ${\displaystyle f(x)}$
  • [if ] against , where so-called denotes the sign function. ${\displaystyle z>n}$ ${\displaystyle \operatorname {sgn}(a_{z})\cdot \operatorname {sgn}(b_{n})\cdot \infty }$
  • [if ] against ${\displaystyle z=n}$ ${\displaystyle {\tfrac {a_{z}}{b_{n}}}}$
  • [if ] against 0 (the x-axis) ${\displaystyle z<n}$
• symmetry
  • If and are both even or both odd, then is even (symmetrical to the y-axis). ${\displaystyle P_{z}}$ ${\displaystyle Q_{n}}$ ${\displaystyle f}$
  • Is even and odd, then is odd (point symmetrical to the origin); The same applies when is odd and even. ${\displaystyle P_{z}}$ ${\displaystyle Q_{n}}$ ${\displaystyle f}$ ${\displaystyle P_{z}}$ ${\displaystyle Q_{n}}$
• Pole: means pole of , if ${\displaystyle x_{p}}$ ${\displaystyle f}$
  • ${\displaystyle Q_{n}(x_{p})=0\quad {\text{und}}\quad P_{z}(x_{p})\neq 0.}$
• Asymptotes: Using polynomial division of by one obtains with polynomials and , where the degree of is smaller than that of . The asymptotic behavior of is thus determined by the completely rational function : ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p=g\cdot q+r}$ ${\displaystyle g}$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle q}$ ${\displaystyle f={\tfrac {p}{q}}=g+{\tfrac {r}{q}}}$ ${\displaystyle g}$
  • ${\displaystyle [z<n]\,}$ x-axis is asymptote: ${\displaystyle g(x)=0}$
  • ${\displaystyle [z=n]\,}$ horizontal asymptote: ${\displaystyle g(x)={\frac {a_{z}}{b_{n}}}}$
  • ${\displaystyle [z=n+1]\,}$ oblique asymptote: ${\displaystyle g(x)=mx+c\,;m\neq 0}$
  • ${\displaystyle [z>n+1]\,}$completely rational approximation function

Integral calculus

Area calculation

The area between the x-axis and the graph of the function f (x) in the interval from a to b is

• ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x,\qquad {\text{falls }}f(x)\geq 0\forall x\in [a,b]}$
• ${\displaystyle -\int _{a}^{b}f(x)\mathrm {d} x,\qquad {\text{falls }}f(x)\leq 0\forall x\in [a,b]}$
• Otherwise the interval must be broken down into such sub-intervals by determining the zeros.

Properties of the definite integral

${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=-\int _{b}^{a}f(x)\mathrm {d} x}$

${\displaystyle \int _{a}^{a}f(x)\mathrm {d} x=0}$

${\displaystyle \int _{a}^{c}f(x)\mathrm {d} x=\int _{a}^{b}f(x)\mathrm {d} x+\int _{b}^{c}f(x)\mathrm {d} x,\qquad a<b<c}$

${\displaystyle \int _{a}^{b}k\cdot f(x)\mathrm {d} x=k\cdot \int _{a}^{b}f(x)\mathrm {d} x}$

${\displaystyle \int _{a}^{b}(f(x)+g(x))\,\mathrm {d} x=\int _{a}^{b}f(x)\mathrm {d} x+\int _{a}^{b}g(x)\mathrm {d} x}$

Integral function and main theorem of differential and integral calculus

Integral function

${\displaystyle F_{a}(x)=\int \limits _{a}^{x}f(t)\mathrm {d} t}$

Law of infinitesimal calculus

${\displaystyle F_{a}(x)'=f(x)\,}$

Indefinite integral

Every function is called an antiderivative of if holds for all x of the domain ${\displaystyle F}$ ${\displaystyle f}$

${\displaystyle F'(x)=f(x)\,}$

This is what the expression means ${\displaystyle \int f(x)\mathrm {d} x}$

integration

If F is any antiderivative of f , then we have

${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=F(b)-F(a)}$

Special antiderivatives

The antiderivatives of are ${\displaystyle f(x)=x^{n}}$

${\displaystyle F(x)={\frac {x^{n+1}}{n+1}}+c,\qquad n\not =-1}$

For everything else, see the table of derivative and antiderivatives

Integration methods

Product, partial or partial integration

• indefinite

  ${\displaystyle \int f(x)g'(x)\mathrm {d} x=f(x)\cdot g(x)-\int f'(x)\cdot g(x)\mathrm {d} x}$

  ${\displaystyle \int f(x)\cdot g(x)\mathrm {d} x=f(x)\cdot G(x)-\int f'(x)\cdot G(x)\mathrm {d} x}$

• certainly

  ${\displaystyle \int _{a}^{b}f(x)\cdot g'(x)\mathrm {d} x=[f(x)\cdot g(x)]_{a}^{b}-\int _{a}^{b}f'(x)\cdot g(x)\mathrm {d} x}$

Integration through substitution

• indefinite

  ${\displaystyle \int f(x)\mathrm {d} x=\int f(\varphi (t))\varphi '(t)\mathrm {d} t}$

• certainly

  ${\displaystyle \int _{a}^{b}f(\varphi (t))\cdot \varphi '(t)\mathrm {d} t=\int _{\varphi (a)}^{\varphi (b)}f(x)\mathrm {d} x}$

• Special case: linear substitution

  ${\displaystyle \int f(mx+n)\mathrm {d} x={\frac {1}{m}}F(mx+n)+C,\qquad m\neq 0}$

  ${\displaystyle \int _{a}^{b}f(mx+n)\mathrm {d} x={\frac {1}{m}}\lbrack F(mx+n)\rbrack _{a}^{b},\qquad m\neq 0}$

• Special case: logarithmic integration

  ${\displaystyle \int {\frac {f'(x)}{f(x)}}\mathrm {d} x=\ln |f(x)|+C,\qquad f(x)\neq 0}$

Applied

Volume determination

• Volume of the body with rotation of the area between the graph of f and the x-axis in the interval [a, b]

  ${\displaystyle \pi \cdot \int _{a}^{b}f^{2}(x)\mathrm {d} x}$

• Volume of the body with rotation of the area between the graph of the reversible function f and the y-axis in the interval [a, b]

  ${\displaystyle \pi \cdot \int _{f(a)}^{f(b)}(f^{-1}(y))^{2}\mathrm {d} y}$

• Volume of the body, which arises from the y-rotation of the area, which is limited by the graph of the function f in the interval [a, b], the x-axis and the two straight lines and ${\displaystyle x=a}$ ${\displaystyle x=b}$

  ${\displaystyle 2\pi \cdot \int _{a}^{b}(x\cdot f(x))\mathrm {d} x}$

Guldinian rules

${\displaystyle M}$ Surface area

${\displaystyle V}$ volume

${\displaystyle L}$ Length of the generating line (profile line)

${\displaystyle A}$ Area of ​​the generating area

${\displaystyle R}$ Radius of the center of gravity

First rule

${\displaystyle M=L\cdot 2\pi R}$

Expressed as a function of the function f (x) of the generating line, this results as:

• with rotation around the x-axis

  ${\displaystyle M=2\pi \int _{a}^{b}f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x.}$

• with rotation around the y-axis

  ${\displaystyle M=2\pi \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}f^{-1}(y){\sqrt {1+\left[\left(f^{-1}(y)\right)'\right]^{2}}}\mathrm {d} y.}$

Second rule

${\displaystyle V=A\cdot 2\pi R.}$

In the case of rotation around the x-axis of a surface between , the x-axis and the limits and the volume is expressed by using as the centroid ${\displaystyle f(x)}$ ${\displaystyle x=a}$ ${\displaystyle x=b}$ ${\displaystyle f(x)}$ ${\displaystyle R}$

${\displaystyle V=A\cdot 2\pi {\tfrac {1}{A}}\int _{A}y\mathrm {d} A=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}$

with and ${\displaystyle y={\tfrac {f(x)}{2}}}$ ${\displaystyle \mathrm {d} A=f(x)\mathrm {d} x.}$

additional

• If f is continuous on [a, b], then the mean value of the function values ​​of f is called ${\displaystyle {\bar {m}}}$ ${\displaystyle [a,b]}$

  ${\displaystyle {\bar {m}}={\frac {1}{b-a}}\cdot \int _{a}^{b}f(x)\mathrm {d} x}$

• Length of the arc of the differentiable function f in the interval [a, b]:

  ${\displaystyle L=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\mathrm {d} x}$

Calculating integrals as an approximation: Numerical integration

• Breakdown sums

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx hf(x_{1})+hf(x_{2})+\cdots +hf(x_{n})\qquad {\text{mit }}h={\frac {b-a}{n}}}$

• Kepler's barrel rule

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {1}{6}}\cdot \left(f(a)+4\cdot f\left({\frac {a+b}{2}}\right)+f(b)\right)}$

• Trapezoidal rule
  • Tendon trapezoid

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {f(b)+f(a)}{2}}\cdot (b-a)}$

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{2n}}\left(f(x_{0})+2f(x_{1})+\cdots +2f(x_{n-1}+f(x_{n})\right)}$

  • Tangent trapezoid

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{2}}\cdot {\frac {b-a}{2}}}$

• Simpson's rule

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{6}}\cdot \left(f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right)}$

  ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{6n}}\cdot \left(f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots f(x_{n})\right)}$

swell

• Konrad Königsberger : Analysis 1 . Springer-Verlag, Berlin et al., 2004, ISBN 3-540-41282-4 .