Pole position

The absolute value of the gamma function goes to infinity at the poles (left). It has no poles on the right and just rises quickly.

In mathematics , a single-point definition gap of a function is called a pole or, for a shorter time, a pole , if the function values in every area around the point (in terms of amount) are arbitrarily large. The poles thus belong to the isolated singularities . The special thing about poles is that the points do not behave chaotically in an environment, but rather strive evenly towards infinity in a certain sense . Therefore, limit value observations can be carried out there.

Generally speaking, one only speaks of poles for smooth or analytical functions . In school mathematics, poles are introduced for real fractional-rational functions . If singularities of other functions, such as transcendent functions , such as in the case of secans , are also to be examined, it is most expedient to consider the analytical continuation on the complex numbers . ${\displaystyle f(x)={\tfrac {1}{\cos x}}}$

Real functions

In the following is a rational function on the real numbers . A more general approach is shown below under Complex Functions . ${\displaystyle f\colon \mathbb {R} \supset X\to \mathbb {R} }$

Poles and gaps in the definition of rational functions that can be continually removed

Every rational function can be written as the quotient of two polynomials :

${\displaystyle f(x)={\frac {A(x)}{B(x)}}={\frac {a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots +a_{1}x+a_{0}}{b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots +b_{1}x+b_{0}}}\quad {\text{mit}}\quad m,n\in \mathbb {N} _{0}}$

Let and be unequal to the zero polynomial . Then poles of can generally only occur at the zeros of the denominator polynomial. So have a -fold zero in . Since zeros can be factored out by means of polynomial division based on the fundamental theorem of algebra , the following applies here: a polynomial of degree and . Now it depends on the numerator polynomial whether there is a pole. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle B}$ ${\displaystyle k}$ ${\displaystyle x_{0}\in \mathbb {R} }$ ${\displaystyle B(x)=(x-x_{0})^{k}\cdot S(x)}$ ${\displaystyle S}$ ${\displaystyle n-k}$ ${\displaystyle S(x_{0})\neq 0}$ ${\displaystyle x_{0}}$

• If true, then is a pole of order . ${\displaystyle A(x_{0})\neq 0}$ ${\displaystyle x_{0}}$ ${\displaystyle k}$
• If there is a -fold zero in , the following applies: ${\displaystyle A}$ ${\displaystyle j}$ ${\displaystyle x_{0}}$
  • if so, then is the pole of with order ; ${\displaystyle j<k}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle k-j}$
  • if so, then there is a definition gap that can be continuously removed and thus not a pole. ${\displaystyle j\geq k}$ ${\displaystyle x_{0}}$

Remarks

• If one interprets “ has in no zero” ${\displaystyle A}$ ${\displaystyle x_{0}}$ as “ has in a -fold zero with ” ${\displaystyle A}$ ${\displaystyle x_{0}}$ ${\displaystyle j}$ ${\displaystyle j=0}$ , then the above distinction can be formulated more briefly.
• Rational functions cannot have any other kind of singularities.
• Rational functions have at most a finite number of poles, since a polynomial can only have a finite number of zeros.

Order of a pole

The order of a pole is expressed by a natural number and is the equivalent of the multiplicity of a zero. The higher the order, the faster the function values ​​tend towards infinity in terms of absolute value. In addition, a distinction is made between even and odd order. Every pole of a rational function has a finite, uniquely determined order. If it is defined as above, then two polynomials are obtained that have no linear factor in common, so that by eliminating all gaps in the definition that can be continuously eliminated. Then has in just then a pole th order if there is a has-fold zero, or in other words, if in a has-fold zero. Likewise, one speaks of a pole of order 0 if it has no pole there. ${\displaystyle f}$ ${\displaystyle P,Q}$ ${\displaystyle f(x)=P(x)/Q(x)}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle k}$ ${\displaystyle Q}$ ${\displaystyle k}$ ${\displaystyle 1/f}$ ${\displaystyle x_{0}}$ ${\displaystyle k}$ ${\displaystyle f}$

Behavior of the graph

f (x) = 1 / x has a first order pole at x = 0

The graph of the function disappears when approaching the pole at infinity and has a vertical asymptote there . The exact behavior is determined by the order of the pole. The higher the order, the steeper the graph appears.

In the case of an odd order, one speaks of a pole with a change in sign , the graph jumps from the positive to the negative image area or vice versa.

In the case of a pole of an even order, the graph lies on both sides of the pole in the image area with the same sign . One then speaks of a pole without a change in sign.

Existence of improper limit values

If there is a pole in , then a limit value only exists if the left-hand and right-hand limit values ​​match. If the order of the pole is straight, this is always given and the limit value is or . ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle \textstyle \lim _{x\to x_{0}}f(x)}$ ${\displaystyle \infty }$ ${\displaystyle -\infty }$

In the case of a pole of odd order, one can only speak of a limit value if it is set. However, this one-point compactification does not contain the less than / equal relation and therefore initially appears unnatural. However, the real numbers can be embedded in the complex numbers and, since these are not arranged, it makes perfect sense. ${\displaystyle +\infty =-\infty }$

Examples

The function has a 2nd order pole at . ${\displaystyle f(x)={\frac {1}{x^{2}}}}$ ${\displaystyle x=0}$

The function has a 3rd order pole at . ${\displaystyle f(x)={\frac {1}{(x-2)^{3}}}}$ ${\displaystyle x=2}$

The function has for a pole of order 2 and for a pole of 1st order. ${\displaystyle f(x)={\frac {x+2}{x^{3}+x^{2}-x-1}}={\frac {(x+2)}{(x+1)^{2}(x-1)}}}$ ${\displaystyle x=-1}$ ${\displaystyle x=1}$

The function has poles of order 1 for and . ${\displaystyle f(x)={\frac {x^{2}+3x+2}{x^{3}+x^{2}-x-1}}={\frac {(x+2)(x+1)}{(x+1)^{2}(x-1)}}={\frac {(x+2)}{(x+1)(x-1)}}}$ ${\displaystyle x=-1}$ ${\displaystyle x=1}$

Difficulty in generalizing

While it does not cause any problems after the above procedure, e.g. As for the tangent function to indicate the existence and order of the pole, it is in the logarithm of impossible. In general, any smooth but non- analytical function causes difficulties. Function-theoretical means offer one possibility to deal with this . ${\displaystyle x=0}$

Complex functions

Let be a domain , a discrete subset, and a holomorphic function. Then on the points of can have three different types of isolated singularities. ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle D\subset G}$ ${\displaystyle f\colon G\backslash D\to \mathbb {C} }$ ${\displaystyle f}$ ${\displaystyle D}$

definition

The following definition contains the poles of real-valued rational functions as a special case. Be . If there is such that in exists, the following cases occur: ${\displaystyle a\in D}$ ${\displaystyle k\in \mathbb {N} _{0}}$ ${\displaystyle \textstyle \lim _{z\to a}(z-a)^{k}\cdot f(z)}$ ${\displaystyle \mathbb {C} }$

• ${\displaystyle k=0}$: Then it can be continued on holomorphic . ${\displaystyle f}$ ${\displaystyle a}$
• ${\displaystyle k\geq 1}$and chosen as small as possible so that the limit value exists. Then there is a pole of order . ${\displaystyle k}$ ${\displaystyle k}$

If there is no such natural number , then has an essential singularity in . ${\displaystyle k}$ ${\displaystyle f}$ ${\displaystyle a}$

From Riemann's theorem of liftability it follows that the limit value already exists if is bounded in a neighborhood of . ${\displaystyle \textstyle \lim _{z\to a}(z-a)^{k}\cdot f(z)}$ ${\displaystyle (z-a)^{k}\cdot f(z)}$ ${\displaystyle a}$

Another characterization of poles is as follows: in has a pole of order if and only if the main part of the Laurent series is finite on a dotted circular disk and the smallest index of a non-vanishing coefficient of the Laurent series is even . ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle k}$ ${\displaystyle a}$ ${\displaystyle -k}$

Meromorphic functions

→ Main article : Meromorphic function

Complex functions that are holomorphic in a domain and whose singularities are at most poles are also called meromorphic. Because of the identity theorem , the set of poles of a meromorphic function can only be discrete. Thus in every compact subset there are at most a finite number of poles. The Mittag-Leffler theorem proves the existence of functions with an infinite number of poles for the entire plane . A divisor can be defined using the order of the poles and zeros of a meromorphic function . ${\displaystyle \mathbb {C} }$

If one considers the compacted closure of the complex numbers , then meromorphic functions map their poles on . If one also allows in the domain of definition, then exactly the polynomials -th degree in have a pole of the order . In general, meromorphic functions are holomorphic if they have at most one pole in. So these are holomorphic functions on a complex manifold , namely the Riemann number sphere . It can be shown that every holomorphic function can be expressed globally as the quotient of two polynomials and is therefore always a rational function . ${\displaystyle {\overline {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$ ${\displaystyle \infty }$ ${\displaystyle {\overline {\mathbb {C} }}}$ ${\displaystyle k}$ ${\displaystyle \infty }$ ${\displaystyle k}$ ${\displaystyle \mathbb {C} \to \mathbb {C} }$ ${\displaystyle {\overline {\mathbb {C} }}}$ ${\displaystyle \infty }$ ${\displaystyle {\overline {\mathbb {C} }}\to {\overline {\mathbb {C} }}}$

Theorem of the integral counting zeros and poles

→ Main article : Residual theorem

Let be a meromorphic function in one domain . Then for every smooth , closed, rectifiable curve that touches neither zero nor poles of and that borders a subset : ${\displaystyle f}$ ${\displaystyle G\subset \mathbb {C} }$ ${\displaystyle \gamma \subset G}$ ${\displaystyle f}$ ${\displaystyle H\subset G}$

${\displaystyle {\frac {1}{2\pi i}}\int _{\gamma }{\frac {f'(z)}{f(z)}}dz=N_{f}-P_{f}}$.

Here and are the number of zero or pole positions including their multiples that lie in. In particular it holds for each on meromorphic function . ${\displaystyle N_{f}}$ ${\displaystyle P_{f}}$ ${\displaystyle H}$ ${\displaystyle G={\overline {\mathbb {C} }}}$ ${\displaystyle N_{f}=-P_{f}}$

Examples

f (x) = 1 / sin (x)

• The function has two poles of the 1st order at . ${\displaystyle f(x)={\tfrac {1}{x^{2}+1}}}$ ${\displaystyle x=\pm i}$

• The reciprocal of the sine can be analytically extended to and has simple odd poles for all integer multiples of π , da . ${\displaystyle f(x)={\tfrac {1}{\sin x}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \textstyle \lim _{z\to n\pi }{\frac {z-n\pi }{\sin z}}=\pm 1}$

• The tangent function has odd poles in all . ${\displaystyle f(z)=\tan z}$ ${\displaystyle z=(k+{\tfrac {1}{2}})\cdot \pi ,\;k\in \mathbb {Z} }$

• The complex logarithm is a superposition and cannot be continued continuously in any field that contains the zero. In it has no pole, but a branch point . ${\displaystyle 0}$

additional

• The pole-zero diagram provides several pieces of information about a signal to be examined .

literature

• Manfred Zimmermann (Ed.): Introductory phase in mathematics. 5th edition. Transparent-Verlag, Berlin 1991, ISBN 3-927055-03-4 .
• Georg Hoever: Higher Mathematics Compact . 2nd, corrected edition. Springer Verlag, Berlin / Heidelberg 2014, ISBN 978-3-662-43994-4 .

Web links

• Zeros, poles, gaps (accessed on August 29, 2016)
• Poles and gaps in the definition of broken rational functions (accessed August 29, 2016)
• Stability and limit stability (accessed on August 29, 2016)