One Has to Analytically Continue o Z o Anticlockwise Round This Singularity

Complex Variable Theory

George B. Arfken , ... Frank E. Harris , in Mathematical Methods for Physicists (Seventh Edition), 2013

Further Applications

An important application of Cauchy's integral formula is the following Cauchy inequality. If $f(z)={\sum }_{}^{}{a}_n{z}^n$ is analytic and bounded, |f (z)| ≤ M on a circle of radius r about the origin, then

(11.37) $|a_n|r^n\le M\text{\hspace{0.17em}(Cauchy's\hspace{0.17em}inequality)}$

gives upper bounds for the coefficients of its Taylor expansion. To prove Eq. (11.37) let us define M(r) = max_|z|=r |f (z)| and use the Cauchy integral for a_n = f ⁽ⁿ⁾(z)/n!,

$|a_n|=\frac{1}{2\pi }\left|\underset{|z|=r}{\oint }\frac{f(z)}{z^{n+1}}dz\right|\le M(r)\frac{2\pi r}{2\pi r^{n+1}}.$

An immediate consequence of the inequality, Eq. (11.37), is Liouville's theorem: If f (z) is analytic and bounded in the entire complex plane it is a constant. In fact, if |f (z)| ≤ M for all z, then Cauchy's inequality Eq. (11.37), applied for |z| = r, gives |a_n | ≤ Mr ⁻ⁿ . If now we choose to let r approach ∞, we may conclude that for all n > 0, |a_n | = 0. Hence f (z) = a ₀.

Conversely, the slightest deviation of an analytic function from a constant value implies that there must be at least one singularity somewhere in the infinite complex plane. Apart from the trivial constant functions then, singularities are a fact of life, and we must learn to live with them. As pointed out when introducing the concept of the point at infinity, even innocuous functions such as f (z) = z have singularities at infinity; we now know that this is a property of every entire function that is not simply a constant. But we shall do more than just tolerate the existence of singularities. In the next section, we show how to expand a function in a Laurent series at a singularity, and we go on to use singularities to develop the powerful and useful calculus of residues in a later section of this chapter.

A famous application of Liouville's theorem yields the fundamental theorem of algebra (due to C. F. Gauss), which says that any polynomial $P(z)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ with n > 0 and a_n ≠ 0 has n roots. To prove this, suppose P(z) has no zero. Then 1/P(z) is analytic and bounded as |z| → ∞, and, because of Liouville's theorem, P(z) would have to be a constant. To resolve this contradiction, it must be the case that P(z) has at least one root λ that we can divide out, forming P(z)/(z − λ), a polynomial of degree n − 1. We can repeat this process until the polynomial has been reduced to degree zero, thereby finding exactly n roots.

Exercises

Unless explicitly stated otherwise, closed contours occurring in these exercises are to be understood as traversed in the mathematically positive (counterclockwise) direction.

11.4.1

Show that

$\frac{1}{2\pi i}{\displaystyle \oint z^{m-n-1}dz,m\text{and}n\text{integers}}$

(with the contour encircling the origin once), is a representation of the Kronecker δ_mn .

11.4.2

Evaluate

$\underset{C}{\overset{}{\oint }}\frac{dz}{z^2-1},$

where C is the circle |z − 1| = 1.

11.4.3

Assuming that f (z) is analytic on and within a closed contour C and that the point z ₀ is within C, show that

$\underset{C}{\overset{}{\oint }}\frac{f^{\prime }(z)}{z-z_0}dz=\underset{C}{\overset{}{\oint }}\frac{f(z)}{{(z-z_0)}^2}dz.$

11.4.4

You know that f (z) is analytic on and within a closed contour C. You suspect that the nth derivative f ⁽ⁿ⁾(z ₀) is given by

$f^{(n)}(z_0)=\frac{n!}{2\pi i}\underset{C}{\overset{}{\oint }}\frac{f(z)}{{(z-z_0)}^{n+1}}dz.$

Using mathematical induction (Section 1.4), prove that this expression is correct.

11.4.5

(a)

A function f (z) is analytic within a closed contour C (and continuous on C). If f (z) ≠ 0 within C and |f (z)| ≤ M on C, show that

$|f(z)|\le M$

for all points within C.

Hint. Consider w(z) = 1/f (z).

(b)

If f (z) = 0 within the contour C, show that the foregoing result does not hold and that it is possible to have |f (z)| = 0 at one or more points in the interior with |f (z)| > 0 over the entire bounding contour. Cite a specific example of an analytic function that behaves this way.

11.4.6

Evaluate

$\underset{C}{\oint }\frac{e^{iz}}{z^3}dz,$

for the contour a square with sides of length a > 1, centered at z = 0.

11.4.7

Evaluate

$\underset{C}{\oint }\frac{{\sin }^{2}z-z^2}{{(z-a)}^3}dz,$

where the contour encircles the point z = a.

11.4.8

Evaluate

$\underset{C}{\oint }\frac{dz}{z(2z+1)},$

for the contour the unit circle.

11.4.9

Evaluate

$\underset{C}{\oint }\frac{f(z)}{z{(2z+1)}^2}dz,$

for the contour the unit circle.

Hint. Make a partial fraction expansion.

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Complex Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

17.2.4 Singular points and Cauchy's residue theorem

17.2.4.1 Types of singularities

Definition 17.7

Consider a function f (z) which is regular (analytic) everywhere on an open set D bounded by a closed contour C (here we are writing C for C ₀), except at a finite number of isolated points $a_1,a_2,\dots ,a_n$ . These exceptional points are called singular points (singularities) of f (z).

Isolated singularities are divided into three types according to the behavior of the function in a deleted neighborhood of the point concerned.

Definition 17.8

An isolated singularity a of the function f (z) is said to be

1.

a removable singularity, if $\underset{z\to a}{\lim }f\left(z\right)$ f (z) exists finitely;

2.

a pole, if $\underset{z\to a}{\lim }f\left(z\right)=\pm \infty$ ;

3.

an essential singularity, f (z) does not tend to a limit (finite or infinite) as $z\to a$ .

Remark 17.3

All of these notions are closely connected with the, so-called, Laurent expansion of the function f (z) which will be discussed below. There will be shown that

•

a removable singularity cannot contain the term $\frac{c}{{\left(z-a\right)}^n}$ for any finite $n\ge 1$ (for example, the function $\frac{\sin \text{undefined}z}{z}$ at the point $z=a=0$ has a removable singularity);

•

evidently, that a function f (z) defined in some deleted neighborhood of z = a has a pole at z = a if and only if the function $g\left(z\right):=\frac{1}{f\left(z\right)}$ is regular at a and has zero at z = a, i.e., g (a) = 0 (while g (z) is not identically equal to zero);

•

in the case of isolated essential singularity there exist (the Sokhotsky–Cazoratti theorem, 1868) at least two sequences $\left\{z_n^{\text{'}}\right\}$ and $\left\{z_n^{″}\right\}$ , each converging to a, such that the corresponding sequences $\left\{f\left(z_n^{\text{'}}\right)\right\}$ and $\left\{f\left(z_n^{″}\right)\right\}$ tend to different limits as $n\to \infty$ (for example, the function $e^{1/z}$ at the point $z=a=0$ has an essential singularity and is regular for all other z).

Definition 17.9

A function f (z) is called meromorphic (ratio type) if its singularities are only poles.

From this definition it immediately follows that in any bounded closed domain of the complex plane a meromorphic function may have only a finite number of poles: for, otherwise, there would exist a sequence of distinct poles converging to a (finite) point in the region; such point would necessarily be a nonisolated singularity that contradicts our hypothesis that any finite singular point of this function must be a pole.

Example 17.4

Meromorphic functions are $1/\sin \text{undefined}z,\tan \text{undefined}z,\cot \text{undefined}z$

17.2.4.2 Cauchy's residue theorem

We enclose the $a_k$ by mutually disjoint circles $C_k$ in D such that each circle $C_k$ enclosing no singular points other than the corresponding point $a_k$ . It follows readily

from (17.25) that the integral of f (z) around $C_k$ is equal to the integral around any other contour $C_k^{\text{'}}$ in D which also encloses a_k , but does not enclose or pass through any other singular point of f (z). Thus the value of this integral is a characteristic of f (z) and the singular point $a_k$

Definition 17.10

The residue of f (z) at the singular point $a_k$ is denoted by $\text{res}f\left(a_k\right)$ and is defined by

(17.26) $\text{res}f\left(a_k\right):=\frac{1}{2\pi i}{\displaystyle \underset{C_k}{\oint }f\left(z\right)dz}$

Formula (17.24) leads immediately to the following result.

Theorem 17.5. (Cauchy's residue theorem)

Let D be an open domain bounded by a closed contour C and let f (z) be regular (analytic) at all points of $\overline{D}$ with the exception of a finite number of singular points $a_1,a_2,\dots ,a_n$ contained in the domain D. Then the integral of f (z) around C is $2\pi i$ times the sum of its residues at the singular points, that is,

(17.27) $\underset{C}{{\displaystyle \oint }}f(z)dz=2\pi i{\displaystyle \sum }_{k=1}^n\text{res}f(a_k)$

Corollary 17.3

The residue of f (z) at a removable singularity is equal to zero.

The next subsection deals with method of residues calculating without integration.

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Magnetics

Ruslan P. Ozerov , Anatoli A. Vorobyev , in Physics for Chemists, 2007

5.1.3 The law of a total current (Ampere law)

The sign of the potential character of a force field is the equality to zero of the circulation of the field intensity vector along any closed contour. Let us see whether the magnetic field is potential, i.e., whether the integral ${\oint }_{\text{L}}\text{B}\cdot dl$ is equal to zero or not.

Consider the simplest case when a magnetic field is created by a linear conductor with current I. The magnetic force lines in this case are the concentric circles lying in parallel planes, perpendicular to the conductor, with their centers on the linear conductor (Figure 5.9). Choose for simplicity contour L coinciding with one circular force line of any radius R (Figure 5.10). Then the circulation of the vector B along the contour L will be equal to

Figure 5.10. An Ampere law consideration.

$\underset{\text{L}}{\oint }\text{B}\cdot dl=\frac{{\mu }_{0}I}{2\pi R}\underset{\text{L}}{\oint }Rd\phi =\frac{{\mu }_{0}I}{2\pi R}\underset{0}{\overset{2\pi R}{\int }}dl.$

Therefore, by using eq. (5.1.19):

(5.1.20) $\underset{\text{L}}{\oint }\text{B}\cdot dl={\mu }_{0}I.$

Since circulation of the magnetic field induction is not zero the magnetic field is not potential. (Notice that in the above integral dl is an element of a contour L but not current.) To obtain this ratio for a noncircular contour of any form is not a difficult task.

Expression (5.1.20) is the essence of Ampere's law: circulation of the induction vector along a closed contour L is equal to the current multiplied by μ ₀ comprised by this contour. (This means that if the current passes outside the contour chosen, this particular current does not contribute to the total current.) A field satisfying the condition (5.1.20) is referred to as a nonpotential field. Expression (5.1.20) is also referred to as Ampere's law to emphasize the unity of the phenomena of interaction of currents with each other and with the magnetic fields.

If contour L comprised N currents then, according to the principle of superposition, the circulation of a vector B is equal to their algebraic sum

(5.1.21) $\underset{\text{L}}{\oint }\text{B}\cdot dl={\mu }_0\sum _{i=1}^{\text{N}}{I}_i,$

the current is considered positive if it corresponds to the clockwise rule, otherwise it is considered negative.

If the current is distributed nonuniformly across the conductor this law can be rewritten as

(5.1.22) $\underset{\text{L}}{\oint }\text{B}\cdot dl={\mu }_0\underset{S}{\int }\text{j}(\text{r})dS,$

where a surface S is resting on contour L (Figure 5.11).

Figure 5.11. A surface rested on a contour loop.

Let us apply Ampere's law to calculate the induction of a magnetic field created by a solenoid. Remember that a coil that has been reeled up by thin wires without misses on the cylinder (Figure 5.12) is referred to as a solenoid. We shall choose a rectangular contour 1–2–3–4, depicted in the figure. Then circulation along the whole contour can be divided into four integrals:

Figure 5.12. Calculation of a solenoid magnetic field strength.

$\underset{\text{L}}{\oint }\text{B}\cdot dl=\underset{\text{L}}{\oint }{B}_{l}dl=\underset{1-2}{\int }{B}_{l}dl+\underset{2-3}{\int }{B}_{l}dl+\underset{3-4}{\int }{B}_{l}dl+\underset{4-1}{\int }{B}_{l}dl.$

It can be seen that the integrals along segments 2–3 and 4–1 are zero since the angle between B and d l is π/2. The integral on the segment 3–4 is also zero because this segment can be chosen far enough from the solenoid where B = 0. Therefore,

$\underset{\text{L}}{\oint }{B}_{l}dl=\underset{1-2}{\int }{B}_{l}dl={\mu }_0\sum _{1-2}{I}_i={\mu }_{0}nlI,$

where l is the length of segment 1–2, n is a number of turns over a unit solenoid length, I is the current in the solenoid. Therefore,

(5.1.23) $B={\mu }_{0}nI,$

i.e., the induction inside an infinitely long solenoid is proportional to the overall current running onto the length unit.

The magnetic field inside the solenoid is uniform. In this respect the solenoid plays the same role as a plate condenser plays in electrostatics.

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Complex Variable Theory

Frank E. Harris , in Mathematics for Physical Science and Engineering, 2014

Integrals on Range $(-\infty \text{,}\infty )$

In contrast to the trigonometric integrals considered earlier, we do not make a change of variables to convert a real integral on $(-\infty \text{,}\infty )$ to a complex integral on a closed contour. Instead, we regard our real integral as a corresponding complex integral along the real axis, and seek to close the contour by an arc which contributes a known (preferably zero) value to the contour integral.

Example 17.6.2 A Simple Integral

We want to evaluate the integral

$I={\int }_0^{\infty }\frac{\mathit{dx}}{1+x^2}=\frac{1}{2}{\int }_{-\infty }^{\infty }\frac{\mathit{dx}}{1+x^2}$

by identifying it as part of a contour integral that can be treated using the residue theorem. As is often the case, our original integral was for the range $(0\text{,}\infty )$ but we use the fact that its integrand is even to extend the range to $(-\infty \text{,}\infty )$ , which is a better starting point for setting up a tractable contour integral.

We therefore consider the contour integral

$J=\frac{1}{2}{\oint }_{C_1}\frac{\mathit{dz}}{1+z^2}$

for the contour shown in Fig. 17.10. The arc in the upper half-plane has radius $R$ , with $R\to \infty$ . The portion of the contour along the real axis therefore corresponds to the integral $I$ , and $I$ and the contour integral $J$ are related by

$J=I+\frac{1}{2}{\int }_{\text{arc at}}R\frac{\mathit{dz}}{1+z^2}\text{.}$

The integral over the arc at radius $R$ is best written in polar coordinates: $z={\mathit{Re}}^{i\theta }$ , with $\mathit{dz}={\mathit{iRe}}^{i\theta }d\theta$ . We get (in the limit of large $R$ )

(17.33) ${\int }_{\text{arc at}}R\frac{\mathit{dz}}{1+z^2}={\int }_0^{\pi }\frac{{\mathit{iRe}}^{i\theta }d\theta }{1+R^2{e}^{2i\theta }}\approx \frac{i}{R}{\int }_0^{\pi }{e}^{-i\theta }d\theta ⟶0\text{.}$

We see that the integrand approaches zero at large $\mid z\mid$ rapidly enough that there is no contribution to the contour integral from the large arc, and we have the simple result that $I=J$ .

Our remaining task is to evaluate the contour integral. Writing

$\frac{1}{1+z^2}=\frac{1}{(z-i)(z+i)}\text{,}$

we see that the integrand of $J$ has two poles, one at $+i$ and the other at $-i$ . The locations of these poles are marked in Fig. 17.10. We note that only the pole at $z=i$ is within the contour; we have

$(\text{residue at}z=i)=\frac{1}{2}{\left[\frac{1}{z+i}\right]}_{z=i}=\frac{1}{2}\frac{1}{2i}=\frac{1}{4i}\text{.}$

Now applying the residue theorem,

(17.34) $I=J=2\pi i\left(\frac{1}{4i}\right)=\frac{\pi }{2}\text{.}$

We could have closed the contour for this example with an arc in the lower half-plane, as shown in Fig. 17.11. The contribution from that arc also vanishes. The contour $C_2$ encloses only the pole at $z=-i$ , and that pole is encircled in the clockwise direction, so its residue should be taken with a minus sign. The reader can verify that when these differences are taken into consideration the lower-arc contour yields (as it must) the same result for the integral $I$ , namely $\pi /2$ .

The integral $I$ is simple enough that it can easily be evaluated directly, thereby checking our foray into complex analysis. We have

${\left.{\int }_0^{\infty }\frac{\mathit{dx}}{1+x^2}={\tan }^{-1}x\right|}_0^{\infty }=\frac{\pi }{2}\text{,}$

in agreement with Eq. (17.34).

The importance of this example, of course, is that the contour-integral evaluation can be applied when elementary integrations are impractical or impossible. $\blacksquare$

Figure 17.10. Contour closed by arc in upper half-plane, for function with poles at $\pm i$ .

Figure 17.11. Contour closed by arc in lower half-plane, for function with poles at $\pm i$ .

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Optical Diffraction

Salvatore Solimeno , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IX.B Watson–Regge Representation

The pth component u ^(p) _d of the scattered field is represented by the series of partial waves indicated in Eq. (31). The summation in Eq. (31) can be considered also as an integration over the complex plane taken around the closed contour C enclosing all the poles due to the zeros of sin (νπ),

$\begin{array}{ll}{\text{u}}_{\text{d}}^{(\text{p})}(\rho ,\theta )& =-\frac{\text{i}}{2}{\int }_{-\infty +\text{i}\varepsilon }^{\infty +\text{i}\varepsilon }\frac{exp\text{i}\left(\frac{1}{2}-\text{p}\right)\pi \nu cos[\nu (\theta +\text{p}\pi )]}{sin\nu \pi }\\ & \times {\text{S}}_{\nu }^{(\text{p})}(\text{k}\text{a}){\text{H}}_{\nu }^{(2)}(\text{k}\rho )\text{d}\nu \end{array}$

To calulate the integral, it is necessary to know the poles of S ^(p) _ν, which can be shown to be close either to the zeros of H ⁽²⁾ _ν(ka), ν _n , or to those of H ⁽¹⁾ _ν(nka),ν′ _n :

$\begin{array}{l}{\nu }_{\text{n}}\simeq -\text{k}\text{a}-{\text{e}}^{\text{i}\pi /3}{\left(\frac{\text{k}\text{a}}{2}\right)}^{1/3}{\text{x}}_{\text{n}}-\frac{\text{i}}{\sqrt{{\text{n}}^2-1}}\\ {\text{v}}_{\text{n}}^{\text{'}}\simeq -\text{m}\text{k}\text{a}+{\text{e}}^{-\text{i}\pi /3}{\left(\frac{\text{m}\text{k}\text{a}}{2}\right)}^{1/3}{\text{x}}_{\text{n}}+\frac{\text{n}}{{({\text{n}}^2-1)}^{1/2}},\end{array}$

where x _n is the zero of Ai(−x).

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Development of the Concept of Homotopy

Ria Vanden Eynde , in History of Topology, 1999

3.1.2. Analysis situs [65].

In 1895 Poincaré digresses on the ideas of the above discussed paper. As an illustration for the functions F _i he considered in 1892, he uses the solutions of a given differential equation with analytic coefficients on the manifold ("variété") V under consideration. V is given by equations f _α(x ₁, x ₂,…, x _n ) = 0 and inequalities φ_β(x ₁,…, x _n ) > 0. In this paper, Poincaré introduces the term "lacet" (loop), p. 240:

If the point M describes an infinitely small contour on the manifold V, the functions F will return to their initial values. This remains true if the point M describes a loop on V, that is to say, if it varies from M ₀ to M ₁ following an arbitrary path M ₀ B M ₁, then describes an infinitely small contour and returns from M ₁ to M ₀ traversing the same path M ₁ B M ₀.

In modern terminology, such a "lacet" is null homotopic. Poincaré uses the notation

$\begin{array}{cc}{M}_{0}B{M}_0\equiv 0& \text{if}{M}_{0}B{M}_0\text{reduces}\text{to}\mathrm{a}\text{loop}.\end{array}$

"To reduce to" is not explained, from what follows we can see that Poincaré interprets this as: along M ₀ B M ₀ all possible functions F _i return to their original value.

The sequence of two paths M ₀ A M ₁ B M ₀ and M₀ B M ₁ C M ₀ is written as M ₀ A M ₁ B M ₀ + M ₀ B M ₁ C M ₀ and the relation M ₀ A M ₁ C M ₀ ≡ M ₀ A M ₁ B M ₀ + M ₀ B M ₁ C M ₀ reflects that all possible functions F _i behave the same way along the two paths in the two members which differ only in a path run through twice in opposite directions. Poincaré emphasizes that the path M ₀ A M ₁ C M ₀ is not the same as the path M ₀ C M ₁ A M ₀ and that the order of the terms in the above sum cannot be changed. The sum notation, tacitly introduced here, later becomes an explicit law of composition for paths [69, p. 523]. The path – E C B F D stands for the path E C B F D traversed in the opposite direction. Since the product of closed paths is not abelian in most manifolds, this sum notation is not very adequate. Compared to Jordan's paper of 1866 we discussed above, and where Jordan uses a product notation, Poincaré's notation constitutes a regression.

In his paper of 1895, Poincaré says (p. 241):

M ₀ B M ₀ ≡ 0, if the closed contour M ₀ B M ₀ constitutes the complete boundary of a 2-dimensional manifold contained in V; and, in fact, this closed contour can then be decomposed into a very large number of loops.

This is not correct, the surface of which M ₀ B M ₀ is the boundary should be simply connected. Poincaré adds (p. 241):

This way we have to take into consideration relations of the form

$k_1{C}_1+k_2{C}_2=k_3{C}_3+k_4{C}_4,$

where the k are integers and the C closed contours drawn on V and starting in M ₀. These relations, which I will call equivalences, resemble the above homologies. They differ from these:

1.

Since, for homologies, the contours can start from an arbitrary initial point;

2.

Since, for homologies, one can change the order of the terms in a sum.

In the fifth complement [68] which we will discuss below, Poincaré reformulates this as (p. 450):

This way for homologies, the terms are composed according to the rules of ordinary addition; for equivalences, the terms are composed according to the same rules as the substitutions in a group; that is why the set of equivalences can be symbolized by a group which is the fundamental group of the manifold.

Even though he emphasizes this difference in working with the homologies and the equivalences there is no difference between his above definition for "M ₀ B M ₀ ≡ 0" and for a "homologie" (p. 207):

Consider a p-dimensional manifold V; let W be a q-dimensional manifold (q ⩽ p) contained in V. Suppose that the boundary of W consists of λ (q − 1)-dimensional manifolds v ₁, v ₂,…, v _λ. We will denote this situation by the notation v ₁ + v ₂ + ··· + v _λ ~ 0.

To me it seems that Poincaré in this one paper uses the term equivalence for two different concepts (homotopy and homology). In [68] the meaning of "K ≡ 0 (mod V)" (K is a closed path (cycle) in the manifold V) is correctly defined as (p. 490):

This means that there is a simply connected region in V, the boundary of which is formed by the cycle K.

For now, the fundamental group of the manifold V is defined as follows [65, p. 242]:

This way, one can imagine a group G satisfying the following conditions:

1.

For each closed contour M ₀ B M ₀ there is a corresponding substitution S of the group.

2.

S reduces to the identical substitution if and only if M ₀ B M ₀ ≡ 0;

3.

If S and S' correspond to the contours C and C' and if C″ = C + C', the substitution corresponding to C″ will be SS'.

The group G will be called the fundamental group of the manifold V.

The second condition will ensure that equivalent paths will lead to the same group element. For Poincaré this was probably intuitively clear. In modern terminology this means that homotopic closed paths lead to the same group element. Poincaré says nothing about the role of the chosen point M ₀ probably because it was clear to him that any other point would lead to the "same" group since the manifolds he considers are arcwise connected. Nowadays, the fundamental group is defined in a more direct way. For two loops α and β based in the same point, the product α · β is the loop obtained by first running along α and then along β. If we go over to the homotopy classes of such loops and define the product of two classes as the class of the product of two representing loops, this product does not depend on the choice of the representing loops and the group structure is ensured. Why Poincaré makes a detour along permutations to define the fundamental group is explained by Hirsch [38, 39]. Using Wussing's [93] results he says that the only groups mathematicians worked with at that time were groups of permutations, as for instance in Galois's and Jordan's papers, or groups of transformations, as in Klein's, Lie's and Jordan's work. As already mentioned earlier this shows why mathematicians did not readily use Cayley's abstract definition of a group (published first in 1854 and again in 1878), a definition to which Cayley himself added the result that every group is in fact a permutation group. To us now it is very easy to recognize a group structure in, for instance, Puiseux's and Jordan's papers, because we see a set of elements with a law of composition. At that time however, such a direct recognition was actually inconceivable because group elements had to operate on something as permutations or transformations do. The "product" is then the law of composition and associativity is ensured. As we shall see below for instance in Tietze's, Dehn's and Gieseking's work, the fundamental group will be introduced by these authors by means of generators and relations. It will gradually lose its characterization as a group of permutations.

After defining the fundamental group Poincaré explains how to calculate it for a given manifold V which is obtained form a polyhedron P ₁ of which the faces are to be identified in pairs in a given manner. Poincaré says that the fundamental group will be derived (this is Jordan's terminology) from a set of principal permutations S _i ("substitutions principales") associated to closed contours C _i which he calls fundamental contours ("contours fermés fondamentaux"). Any other closed contour will be equivalent to a combination of these fundamental contours. The fundamental contours may satisfy a relation of the form $k_2{C}_2+{k^{\prime }}_1{C}_1+k_3{C}_3\equiv 0$ which Poincaré interprets as follows, p. 243:

This means that the substitution $S_1^{k_1}{S}_2^{k_2}{S}_1^{k^{\prime }}{S}_3^{k_3}$ reduces to the identical substitution.

It is clear that we obtain the fundamental contours as follows. Let M ₀ be a point interior to P ₁, A a point on one of the faces of P ₁, A' the corresponding point on the conjugated face. One will pass from M ₀ to A, then from A' to M ₀ without leaving P ₁; the corresponding path on the manifold V will be closed. This way there are as many fundamental contours as there are pairs of faces. In order to form the fundamental equivalences: Consider a cycle of edges. Let, for instance, an edge be the intersection of the faces F ₁ and F'_μ, which I therefore will call the edge F ₁ F'_μ; let F'₁ be the conjugated face of F ₁ and F ₂ F'₁ the conjugated edge of F ₁ F'_μ on this face; let F'₂ be the conjugated face of F ₂ and F ₃ F'₂ the conjugated edge of F ₂ F'₁ on this face; etc. until we return to the face F'_μ and the edge F ₁ F'_μ. Note that while performing this operation we can return, several times, to the same face. Let A _i be a point of F _i and let A' _i be the corresponding point on F' _i ; let C _i be the fundamental contour M ₀ A _i + A' _i M ₀. We will have the fundamental equivalence C ₁ + C ₂ + ··· + C _μ ≡ 0. This way there will be as many fundamental equivalences as there are cycles of edges. Once we have formed the fundamental equivalences in this way, we can deduce the fundamental homologies differing from them by the fact that the order of the terms is irrelevant. From these homologies the determination of the Betti number P _i will follow.

As follows from the above, Poincaré faulty definition of the equivalence M ₀ B M ₀ ≡ 0 does not affect his calculation because he uses another criterion to decide whether a group element is the identity. It is clear that Poincaré got his inspiration from his work on automorphic functions (see [65, p. 247]; [61,62]). In modern terminology he works with closed paths in the universal covering space. These correspond to null homotopic closed paths in the underlying space. Poincaré inattention may even be the result of his method since in the universal covering space any closed curve is both null homotopic and null homologous.

Poincaré illustrates his method by examples. These show that Poincaré works with what we now call homology with rational coefficients and this implies that at this stage the torsion coefficients escape his notice [4]. As Bollinger points out, Poincaré says in the second complement [67, p. 339]: "We will combine … the homologies using addition, subtraction, multiplication and sometimes division." Thus Poincaré recognizes the difference between homology (with coefficients in ℤ) and homology with rational coefficients. After a comment made by P. Heegaard on the duality theorem [65], Poincaré also points out that his definition of the Betti numbers differs from the one Betti gave. In modern terminology, whereas Betti works with homology mod 2, Poincaré uses homology with coefficients in ℤ. One of the examples is a 3-manifold with non-trivial fundamental group but with the same Betti numbers as the 3-sphere, which causes Poincaré to redefine the term simple connectivity. Whereas Riemann and Betti had defined connectivity in terms of boundaries (i.e. for us homology) and simple connected manifolds as manifolds for which all connectivity numbers are 1, Poincaré reserves the term simply connected for manifolds with trivial fundamental group.

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GENERATING ISOVALUE CONTOURS FROM A PIXMAP

Tim Feldman , in Graphics Gems III (IBM Version), 1992

Publisher Summary

This chapter presents an algorithm that follows the edge of a contour in an array of sampled data. It discusses Freeman chain encoding to produce a list of vectors that describe the outline of the contour. The algorithm is capable of handling contours containing a single sample point, contours surrounding regions of a different elevation, contours that do not form closed curves, and contours that form curves that cross themselves, forming loops. In all cases, it follows the outermost edge of the contour. Given an initial point in an elevation contour in the array, the algorithm finds the edge of the contour. Then it follows the edge in a clockwise direction until it returns to its starting point. The vectors may be thought of as the direction part of a traditional two-dimensional vector, with the length part always equal to one pixel. Once the number of vectors is determined, the memory is allocated, and the main algorithm is called to retrace the contour and pack the direction vectors into the memory block.

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Introduction and Preliminaries

H.M. Srivastava , Junesang Choi , in Zeta and q-Zeta Functions and Associated Series and Integrals, 2012

From an Application of the Residue Calculus

Consider a function $f\left(z\right)$ given by

$\begin{array}{lll}\hfill f\left(z\right)=\frac{1}{z}\left(\frac{1}{1+z^n}-e^{i{z}^n}\right)\left(n\mathfrak{\in }\mathbb{N}\right).& & \hfill \end{array}$

Since

$\underset{z\to 0}{lim}f\left(z\right)=\left\{\begin{array}{cc}\hfill -1-i& \left(n=1\right)\hfill \\ \hfill \\ \hfill 0& \left(n\mathfrak{\in }\mathbb{N}\mathfrak{\setminus }\mathfrak{\{}1\mathfrak{\}}\right),\hfill \end{array}\right.$

the function $f\left(z\right)$ has a removable singularity at z = 0 and simple poles at

$\begin{array}{lll}\hfill z=\exp \left(\frac{\left(2k+1\right)\pi i}{n}\right)\left(k=0,1,\dots ,n-1\right).& & \hfill \end{array}$

We now consider a counterclockwise-oriented simple closed contour:

$\begin{array}{lll}\hfill C:=C_{\delta }\cup L_1\cup C_R\cup L_2\left(0<\delta <1<R\right),& & \hfill \end{array}$

where

$\begin{array}{lll}\hfill C_{\delta }:z=\delta e^{i\theta }\left(\theta \text{\hspace{0.17em}varies}\text{\hspace{0.17em}from}\frac{\pi }{2n}\text{\hspace{0.17em}to}0\right),& & \hfill \end{array}$

L ₁ a line segment from δ to R on the positive real axis,

$\begin{array}{lll}\hfill C_R:z=R{e}^{i\theta }\left(\theta \text{\hspace{0.17em}varies}\text{\hspace{0.17em}from}0\text{\hspace{0.17em}to}\frac{\pi }{2n}\right)& & \hfill \end{array}$

and

$\begin{array}{lll}\hfill L_2:z=x\exp \left(\frac{i\pi }{2n}\right)\left(x\text{\hspace{0.17em}varies}\text{\hspace{0.17em}from}R\text{\hspace{0.17em}to}\delta \right),& & \hfill \end{array}$

that is, a line segment on the half-line beginning at the origin with the argument $\frac{\pi }{2n}$ . Since $f\left(z\right)$ is analytic throughout the domain interior to and on the closed contour C, it follows from the Cauchy-Goursat theorem that

$\begin{array}{lll}\hfill \left(\underset{C_{\delta }}{\int }+\underset{L_1}{\int }+\underset{C_R}{\int }+\underset{L_2}{\int }\right)f\left(z\right)dz=0,& & \hfill \end{array}$

which, upon taking the limits as

$\begin{array}{lll}\hfill \delta \to 0+\text{\hspace{0.17em}and}R\to \mathfrak{\infty }& & \hfill \end{array}$

and equating the real and imaginary parts of the last resulting equation, yields the following two interesting integral identities:

(71) $\begin{array}{lll}\hfill \underset{0}{\overset{\mathfrak{\infty }}{\int }}\left(\frac{1}{1+x^n}-\cos \left(x^n\right)\right)\frac{dx}{x}=\underset{0}{\overset{\mathfrak{\infty }}{\int }}\left(\frac{1}{1+x^{2n}}-\exp \left(-x^n\right)\right)\frac{dx}{x}\left(n\mathfrak{\in }\mathbb{N}\right)& & \hfill \end{array}$

and

(72) $\begin{array}{ll}\hfill \underset{0}{\overset{\mathfrak{\infty }}{\int }}\frac{x^{n-1}}{1+x^{2n}}dx=\underset{0}{\overset{\mathfrak{\infty }}{\int }}\frac{\sin \left(x^n\right)}{x}dx=\frac{\pi }{2n}\left(n\mathfrak{\in }\mathbb{N}\right).& \end{array}$

It is noted that the integral identity (71) is a special case of (52) or (53). Moreover, (72) can be evaluated, as above, by applying the residue calculus to another function

$\begin{array}{lll}\hfill \mathfrak{f}\left(z\right)=\frac{\exp \left(i{z}^n\right)}{z}\left(n\mathfrak{\in }\mathbb{N}\right)& & \hfill \end{array}$

and a counterclockwise-oriented simple closed contour

$\begin{array}{lll}\hfill \mathcal{C}:=C_{\delta }\cup L_1\cup C_R\cup L_2\left(0<\delta <R\right),& & \hfill \end{array}$

where $C_{\delta }$ and L ₁ are the same as above,

$\begin{array}{lll}\hfill C_R:z=R{e}^{i\theta }\left(\theta \text{\hspace{0.17em}varies}\text{\hspace{0.17em}from}0\text{\hspace{0.17em}to}\frac{\pi }{n}\right)& & \hfill \end{array}$

and

$\begin{array}{lll}\hfill L_2:z=x\exp \left(\frac{i\pi }{n}\right)\left(x\text{\hspace{0.17em}varies}\text{\hspace{0.17em}from}R\text{\hspace{0.17em}to}\delta \right).& & \hfill \end{array}$

We conclude this section by remarking that more integral representations for γ can be obtained by applying the same techniques employed here (see [302]) or other methods (if any) to some other known formulas that have not been used (see [572]).

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Cardinal L-splines according to Micchelli

Ognyan Kounchev , in Multivariate Polysplines, 2001

Corollary 13.24

For every λ such that λ ≠ e ^{λ _j} for all j = 1, 2, …, Z + 1, the function A_Z (x; λ) permits the residuum representation

(13.11) $A_Z\left(x;\lambda \right)=\frac{1}{2\pi i}{\displaystyle {\int }_{\Gamma }\frac{1}{qz+1\left(z\right)}\frac{e^{xz}}{e^z-\lambda }dz,}$

where the closed contour Γ surrounds the zeros of q _Z+1(z), i.e. all elements of Λ = [λ₁, λ₂, …, λ _Z+1], and excludes the zeros of the function exp(xz)/(e^z – λ).

The proof is due to the Frobenius representation of divided difference as residuum, see formula (11.12), p. 193.

Let us put

(13.12) $r\left(\lambda \right):={\displaystyle \prod _{j=1}^{Z+1}\left(e^{{\lambda }_j}-\lambda \right)={\displaystyle \sum _{j=0}^{Z+1}{r}_j{\lambda }^j,}}$

(13.13) $s\left(\lambda \right):={\displaystyle \prod _{j=1}^{Z+1}\left(e^{-{\lambda }_j}-\lambda \right)}={\displaystyle \sum _{j=0}^{Z+1}{s}_j{\lambda }^j.}$

We have the following important representation of the function A_Z (x; λ).

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Theory of Intense Beams of Charged Particles

Valeriy A. Syrovoy , in Advances in Imaging and Electron Physics, 2011

4.4.2 Representation of z ^ν in the Form of a Contour Integral

We use the Laplace transformation with the fractional positive power index

(4.24) $z^{\mathrm{\nu }}=\frac{1}{2\mathrm{\pi }i}\underset{a-i\infty }{\overset{a+i\infty }{\int }}B\left(p\right)e^{\mathit{pz}}dp,\mathrm{\nu }>0;B\left(p\right)=\underset{0}{\overset{\infty }{\int }}{z}^{\mathrm{\nu }}{e}^{-pz}dz.$

Here the first integral is taken along the straight line Re p  = a  >   0 from the bottom to top; B(p) is the power function image. Let p  = re ^iα (−   π/2   <   α   <   π/2) be any complex number belonging to the right half-plane Rep  >   0. Let us introduce the new integration variable t  = pz in the integral for B(p). At p fixed, the integration contour turns into the ray arg t  =   α, which we denote as L:

(4.25) $B\left(p\right)=p^{-\mathrm{\nu }-1}\underset{L}{\int }{t}^{\mathrm{\nu }}{e}^{-t}dt.$

Consider a closed contour ℒ in the complex plane t (Figure 26), which confines a sector within a circle of the radius R and involves a segment of the real axis, the radius L _R , and the circular arc C _R : $t=R{e}^{\mathit{i\vartheta }}$ , 0   <   ϑ   <   α. Since the integrand in Eq. (4.25) is regular inside the sector, the integral taken along the closed contour ℒ is zero in accordance with the Cauchy theorem. Let us estimate the integral over C _R :

Figure 26. Integration contour in the complex plane t.

(4.26) $I=\left|\underset{C_R}{\int }{t}^{\mathrm{\nu }}{e}^{-t}dt\right|=R^{\mathrm{\nu }+1}\left|\underset{0}{\overset{\mathrm{\alpha }}{\int }}{e}^{-Rcos\mathrm{\vartheta }}expi\left[\left(\mathrm{\nu }+1\right)\mathrm{\vartheta }+\frac{\mathrm{\pi }}{2}-Rsin\mathrm{\vartheta }\right]d\mathrm{\vartheta }\right|.$

The module of the second exponent under the integral does not exceed unity; cos   ϑ   ≥   ε   >   0 at −   π/2   <   α   <   π/2. Thus, the integral over C _R can be majorized by the expression I  ≤ R ^{ν   +   1}|e ^{−   εR} α|   →   0 at R  →   0.

Passing to the limit and taking into consideration that the direction of the integration along the contour ℒ is inverse compared with that in Eq. (4.25), we can see that the contour L, which L _R is approaching, may be replaced by the real axis

(4.27) $B\left(p\right)=p^{-\mathrm{\nu }-1}\underset{0}{\overset{\infty }{\int }}{t}^{\mathrm{\nu }}{e}^{-t}dt=p^{-\mathrm{\nu }-1}\mathrm{\Gamma }\left(\mathrm{\nu }+1\right).$

The image of the power function z ^ν is thus expressed through the Γ-function. Let us now consider the integral along the closed contour ℒ shown in (Figure 27a):

Figure 27. Integration contour in the complex plane p, with circumvention of the singularity at the coordinate origin.

(4.28) $I=\frac{1}{2\mathrm{\pi }i}\underset{ℒ}{\int }B\left(p\right)e^{\mathit{pz}}dp.$

The function (4.27) is regular inside ℒ (we have circumvented the singularity at the coordinate origin); therefore, I  =   0. At R  → ∞ we have B(p)   →   0; the same, according to the Jordan lemma, occurs with the integral along C _R at b  → ∞. From I  =   0 and Eqs. (4.24) we have (Figure 27b):

(4.29) $z^{\mathrm{\nu }}=\frac{\mathrm{\Gamma }\left(\mathrm{\nu }+1\right)}{2\mathrm{\pi }i}\underset{C}{\int }{p}^{-\mathrm{\nu }-1}{e}^{\mathit{pz}}dp.$

Putting $\overline{p}=-p$ and taking into account that (−   1)^{ν   +   1}  = e ^{iπ(ν   +   1)}, we come to the expression (Figure 27c):

(4.30) $z^{\mathrm{\nu }}=-\frac{\mathrm{\Gamma }\left(\mathrm{\nu }+1\right)}{2\mathrm{\pi }i}{e}^{i\mathrm{\pi }(\mathrm{\nu }+1)}\underset{\mathfrak{c}}{\int }{\overline{p}}^{-\mathrm{\nu }-1}{e}^{-\overline{p}z}d\overline{p}.$

Using the well-known property of the Γ-function, Γ(ν   +   1)Γ(−   ν)×sin[(ν +   1)π]   =   π, we obtain

(4.31) $z^{\mathrm{\nu }}=-\frac{\mathrm{\pi }exp\left[i\mathrm{\pi }\left(\mathrm{\nu }+1\right)\right]}{2\mathrm{\pi }isin\left[\left(\mathrm{\nu }+1\right)\mathrm{\pi }\right]\mathrm{\Gamma }\left(-\mathrm{\nu }\right)}\underset{\mathfrak{c}}{\int }{p}^{-\mathrm{\nu }-1}{e}^{-pz}dp=\frac{1}{\mathrm{\Gamma }\left(-\mathrm{\nu }\right)}\underset{\left(0\right)}{\overset{\infty }{\int }}{p}^{-\mathrm{\nu }-1}{e}^{-pz}dp.$

We again use the notation p for the integration variable in Eq. (4.31), while the symbol (0) in the lower limit of the second integral (which, in contrast to the integral over $\mathfrak{c}$ , represents a real value) indicates that the singularity is circumvented in the course of integration.

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