# Complex functions

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Are you struggling to understand the Complex Functions unit? Then you may be missing the following basics:

### Features - Basics20 min.

#### MathematicsFunctionsFunctions

A function assigns the value of a variable x, designated as independent, to exactly one value of a second variable y, the dependent. This unambiguous assignment of a value is also referred to as mapping the set of x values ​​to a set of y values. The number of independent variables can also be higher than one, but not that of dependent variables. In the abstract form the function is a relation to two sets. Explanation of the concept of function, the types of representation of functions and elementary examples are the subject of this learning unit.

### Introduction to Complex Numbers20 min.

#### MathematicsCountingComplex numbers

A complex number is an extension of a real number. The term complex number is defined and the connection of complex numbers by addition, subtraction, multiplication and division is explained.

## Physics, chemistry and biology with single molecules (finished)

The technological developments of the last decade - especially in the field of microelectronics - are characterized by a rapidly advancing process of miniaturization. With the ever smaller dimensions of components, however, the limits of conventional microstructuring are emerging, so there is a need for fundamentally new concepts. One of the approaches to this is based on the production of nanostructures using a modular principle with functionalized molecules. In parallel, and so far largely independently of this, advances have been made in supramolecular chemistry, which deals with highly complex molecular structures, with the physical, chemical and biological properties of these structures going far beyond those of the subunits. Improved experimental techniques from recent times - such as scanning probe methods or spatially resolved laser fluorescence spectroscopy - now open up the possibility of examining individual molecules and tailoring their function for applications, for example in nanotechnology, or also to clarify their effect in biochemical reaction chains. When using the technically demanding methods for the detection of single molecules in basic research, new results are expected primarily from interdisciplinary cooperation.

With this focus, the foundation would therefore like to encourage natural scientists to examine single molecules as individual entities in their nanoscale environment and to develop new methods for their direct control. The focus is on uncovering single molecule-specific phenomena and thus creating the scientific basis for applications. Both biologically relevant macromolecules and synthetically produced molecules can be considered as molecular building blocks. The characterization and manipulation of the individual molecules can relate to mechanical, electrical, optical, chemical or biological properties. Starting from the single molecule, differences to ensemble mean values ​​as well as the assembly and interaction of molecular building blocks to achieve new functions can be investigated.

I. Motivation

Single molecule measurement methods are characterized by the fact that they can be used to reveal specific effects due to fluctuations in time or local inhomogeneities that remain hidden in classical measurements on ensembles of molecules. With its focus, established in 1997, the Volkswagen Foundation would like to encourage the use of this unique possibility of single-molecule methods in order to understand processes at the molecular level in more detail. A very promising interdisciplinary field of work is emerging here for basic research, which deals with the properties of single molecules, their changes due to external influences and potential applications in molecular nanotechnology. There is a need for research in the targeted methodological further development of single-molecule and also single-particle techniques, in the testing of single-molecule manipulation for the direct control of molecular building blocks and complexes made up of them, as well as in the elucidation of single events in (bio) chemical reaction chains.

II. Objective

It is aimed at natural scientists who examine single molecules as individual entities in their nanoscale environment or who would like to develop new methods for their characterization or manipulation. The focus aims to uncover single-molecule-specific phenomena. Scientific questions are to be formulated and processed that cannot be answered satisfactorily with conventional averaging measurement methods. Ultimately, it is also important to create the scientific basis for applications - for example in cell biology or nanotechnology. The characterization and manipulation of the individual molecules or individual particles can relate to mechanical, electrical or spectroscopic properties and also to chemical or biological functions.

The focus on single molecules is the central criterion for the assignment to the focus. In addition, projects related to individual particles and individual nano-objects are also taken into account, which aim at recording individual measured variables that cannot be derived from ensemble mean values. Experimental and theoretical approaches (including computer simulation) can be pursued. The spectrum of questions to be dealt with can range from fundamental quantum mechanical or spectroscopic investigations to the targeted assembly of molecular building blocks through covalent or non-covalent interactions to achieve new functions. The subject of research should be suitably prepared, for example complex molecules, supramolecular units or nano-objects fixed on surfaces or in a matrix. Contributions to the relationship between the properties of individual molecules and a macroscopic thermodynamic description can also be sought with the investigations.

III. Demarcation

Work on individual atoms, ions, photons and macroscopic solids does not fall under the objective. Only applications in which the use of single-molecule methods is absolutely necessary will be considered. Investigations on ensembles of isolated molecules as well as molecular biological or biochemical work are only considered in the context of the selected single molecule approach. Purely empirical-technological further developments, for example in the area of ​​single molecule detection or the so-called optical tweezers (laser tweezers), are not the subject of the focus.

IV. Funding opportunities

Research projects are funded through the provision of personnel and material resources including travel allowances, also for stays of employees within the funded projects with cooperation partners, possibly also abroad, for limited periods of time (maximum one year). Cooperation projects involving various fields of physics, chemistry and biology as well as coordinated experimental-theoretical projects are particularly desirable. As a rule, permits are initially issued for a period of up to three years. As long as the focus remains, the project can be continued after a renewed application and technical examination (including the work results available up to that point) for a total funding period of five years.

Regular focus symposia are planned to intensify the interdisciplinary exchange of experiences and results, especially among the working groups funded. In addition, scientific events can be funded in accordance with the modalities of the "Symposia and Summer Schools" program.

V. Application

Applications must be sent in writing (in English) and in duplicate to the Volkswagen Foundation's office. There are no application forms.

In principle, applications from abroad are treated the same as those from Germany, but the foundation expects project-related cooperation with academic institutions in Germany, the necessity of which is based on the specific situation of the project.

## Application and meaning

### Biological importance

Complexes also play an important role in biology. These can be catalytically active proteins (enzymes) or catalytically inactive proteins. Many enzymes contain complexes in their active centers. This topic is one of the key areas of bioinorganic chemistry. In general, a complexing metal atom is present here, which is not completely complexed by amino acid side chains as ligands. A ligand site acts as an active center for the conversion or temporary binding of the substrate. The most common complex centers are iron, copper, zinc, calcium, magnesium and manganese. But there are also more unusual elements such as vanadium. Calcium in particular, as well as zinc complexes, are of structural importance (e.g. zinc fingers in DNA sequence recognition). The catalytically inactive proteins include z. B. Porphyrin complexes such as the heme in hemoglobin and in cytochromes, or chlorophyll (chelate complexes in each case). See also:

### Complexing agents

Various Complexing agents serve as food additives:

(E 385) (E 574)
• Isoascorbic acid (E 315)
• Sodium isoascorbate (E 316) (E 432)
• Polysorbate 40 (E 434)
• Polysorbate 60 (E 435) (E 433) (E 334) (E 330)
(E 331) (E 332) (E 333)
• (This is particularly biodegradable Complexing agents is used in water cycles to prevent and dissolve limescale deposits.)

In analytical chemistry, complex formation reactions with certain complexing agents are important as detection reactions (for copper, silver, nitrate / ring samples, bismuth ions). See also chelatometry.

## Complex functions, simple operation

Functions that cannot be used efficiently due to poor product design reduce sales opportunities in the medium term.

User-friendly product design is therefore now part of the development strategy in many companies. The term Useware describes the design of the hardware and software components involved in the use. Useware engineering aims to analyze, design and implement systems in a user-friendly manner using scientific methods.

Solutions for vehicle, process and production technology, for engineering and for information presentation as well as interaction concepts will be presented at the USEWARE conference. The center for human-machine interaction at the German Research Center for Artificial Intelligence in Kaiserslautern supports the event. The 4th USEWARE conference on October 15 and 16, 2008 in Baden-Baden will be organized by the VDI Wissensforum.

One focus of the conference is on useware projects in vehicle technology. Among other things, the interaction with large amounts of data in automobiles, practicable dimensions for HMI evaluation, touchscreen operation in the vehicle and the measurement of cognitive driver distraction will be discussed. New findings on the subject of useware engineering are also deepened.

In this section, the lectures deal, for example, with eye movement analysis for software ergonomic evaluation, the analysis of cognitive user models (SimTrA) and a workflow-based human-machine evaluation system using digital cameras as an example. Other topics that engineers, computer scientists, psychologists and designers will be discussing at USEWARE are alarm management and human-process communication as well as human-computer interaction design.

## Holomorphic functions

holomorphic functions, complex-valued functions f(z) that are differentiable. A feature that works in one field Γ is holomorphic, always fulfills the Cauchy-Riemann differential equation (Function theory).

Staff Volume I and II

Silvia Barnert
Dr. Matthias Delbrück
Dr. Reinald ice cream
Natalie Fischer
Walter Greulich (editor)
Carsten Heinisch
Sonja Nagel
MS (Optics) Lynn Schilling-Benz
Dr. Joachim Schüller

Christine Weber
Ulrich Kilian

The author's abbreviation is in square brackets, the number in round brackets is the subject area number, a list of subject areas can be found in the foreword.

Katja Bammel, Berlin [KB2] (A) (13)
Prof. Dr. W. Bauhofer, Hamburg (B) (20, 22)
Sabine Baumann, Heidelberg [SB] (A) (26)
Dr. Günther Beikert, Viernheim [GB1] (A) (04, 10, 25)
Prof. Dr. Hans Berckhemer, Frankfurt [HB1] (A, B) (29)
Prof. Dr. Klaus Bethge, Frankfurt (B) (18)
Prof. Tamás S. Biró, Budapest [TB2] (A) (15)
Dr. Thomas Bührke, Leimen [TB] (A) (32)
Angela Burchard, Geneva [AB] (A) (20, 22)
Dr. Matthias Delbrück, Dossenheim [MD] (A) (12, 24, 29)
Dr. Wolfgang Eisenberg, Leipzig [WE] (A) (15)
Dr. Frank Eisenhaber, Heidelberg [FE] (A) (27 Essay Biophysics)
Dr. Roger Erb, Kassel [RE1] (A) (33)
Dr. Angelika Fallert-Müller, Groß-Zimmer [AFM] (A) (16, 26)
Dr. Andreas Faulstich, Oberkochen [AF4] (A) (Essay Adaptive Optics)
Prof. Dr. Rudolf Feile, Darmstadt (B) (20, 22)
Stephan Fichtner, Dossenheim [SF] (A) (31)
Dr. Thomas Filk, Freiburg [TF3] (A) (10, 15)
Natalie Fischer, Dossenheim [NF] (A) (32)
Prof. Dr. Klaus Fredenhagen, Hamburg [KF2] (A) (Essay Algebraic Quantum Field Theory)
Thomas Fuhrmann, Heidelberg [TF1] (A) (14)
Christian Fulda, Heidelberg [CF] (A) (07)
Frank Gabler, Frankfurt [FG1] (A) (22 essay data processing systems for future high-energy and heavy-ion experiments)
Dr. Harald Genz, Darmstadt [HG1] (A) (18)
Michael Gerding, Kühlungsborn [MG2] (A) (13)
Andrea Greiner, Heidelberg [AG1] (A) (06)
Uwe Grigoleit, Göttingen [UG] (A) (13)
Prof. Dr. Michael Grodzicki, Salzburg [MG1] (A, B) (01, 16 essay density functional theory)
Prof. Dr. Hellmut Haberland, Freiburg [HH4] (A) (Essay Cluster Physics)
Dr. Andreas Heilmann, Chemnitz [AH1] (A) (20, 21)
Carsten Heinisch, Kaiserslautern [CH] (A) (03)
Dr. Hermann Hinsch, Heidelberg [HH2] (A) (22)
Jens Hoerner, Hanover [JH] (A) (20)
Dr. Dieter Hoffmann, Berlin [DH2] (A, B) (02)
Renate Jerecic, Heidelberg [RJ] (A) (28)
Dr. Ulrich Kilian, Hamburg [UK] (A) (19)
Thomas Kluge, Mainz [TK] (A) (20)
Achim Knoll, Strasbourg [AK1] (A) (20)
Andreas Kohlmann, Heidelberg [AK2] (A) (29)
Dr. Barbara Kopff, Heidelberg [BK2] (A) (26)
Dr. Bernd Krause, Karlsruhe [BK1] (A) (19)
Ralph Kühnle, Heidelberg [RK1] (A) (05)
Dr. Andreas Markwitz, Dresden [AM1] (A) (21)
Holger Mathiszik, Bensheim [HM3] (A) (29)
Mathias Mertens, Mainz [MM1] (A) (15)
Dr. Dirk Metzger, Mannheim [DM] (A) (07)
Dr. Rudi Michalak, Warwick, UK [RM1] (A) (23)
Helmut Milde, Dresden [HM1] (A) (09 Essay Acoustics)
Guenter Milde, Dresden [GM1] (A) (12)
Maritha Milde, Dresden [MM2] (A) (12)
Dr. Christopher Monroe, Boulder, USA [CM] (A) (Essay Atom and Ion Traps)
Dr. Andreas Müller, Kiel [AM2] (A) (33 Essay Everyday Physics)
Dr. Nikolaus Nestle, Regensburg [NN] (A) (05)
Dr. Thomas Otto, Geneva [TO] (A) (06 Essay Analytical Mechanics)
Prof. Dr. Harry Paul, Berlin [HP] (A) (13)
Cand. Phys. Christof Pflumm, Karlsruhe [CP] (A) (06, 08)
Prof. Dr. Ulrich Platt, Heidelberg [UP] (A) (Essay Atmosphere)
Dr. Oliver Probst, Monterrey, Mexico [OP] (A) (30)
Dr. Roland Andreas Puntigam, Munich [RAP] (A) (14 Essay General Theory of Relativity)
Dr. Gunnar Radons, Mannheim [GR1] (A) (01, 02, 32)
Prof. Dr. Günter Radons, Stuttgart [GR2] (A) (11)
Oliver Rattunde, Freiburg [OR2] (A) (16 essay cluster physics)
Dr. Karl-Henning Rehren, Göttingen [KHR] (A) (Essay Algebraic Quantum Field Theory)
Ingrid Reiser, Manhattan, USA [IR] (A) (16)
Dr. Uwe Renner, Leipzig [UR] (A) (10)
Dr. Ursula Resch-Esser, Berlin [URE] (A) (21)
Prof. Dr. Hermann Rietschel, Karlsruhe [HR1] (A, B) (23)
Dr. Peter Oliver Roll, Mainz [OR1] (A, B) (04, 15 essay distributions)
Hans-Jörg Rutsch, Heidelberg [HJR] (A) (29)
Dr. Margit Sarstedt, Newcastle upon Tyne, UK [MS2] (A) (25)
Rolf Sauermost, Waldkirch [RS1] (A) (02)
Prof. Dr. Arthur Scharmann, Giessen (B) (06, 20)
Dr. Arne Schirrmacher, Munich [AS5] (A) (02)
Christina Schmitt, Freiburg [CS] (A) (16)
Cand. Phys. Jörg Schuler, Karlsruhe [JS1] (A) (06, 08)
Dr. Joachim Schüller, Mainz [JS2] (A) (10 essay analytical mechanics)
Prof. Dr. Heinz-Georg Schuster, Kiel [HGS] (A, B) (11 essay Chaos)
Richard Schwalbach, Mainz [RS2] (A) (17)
Prof. Dr. Klaus Stierstadt, Munich [KS] (A, B) (07, 20)
Cornelius Suchy, Brussels [CS2] (A) (20)
William J. Thompson, Chapel Hill, USA [WYD] (A) (Essay Computers in Physics)
Dr. Thomas Volkmann, Cologne [TV] (A) (20)
Dipl.-Geophys. Rolf vom Stein, Cologne [RVS] (A) (29)
Patrick Voss-de Haan, Mainz [PVDH] (A) (17)
Thomas Wagner, Heidelberg [TW2] (A) (29 essay atmosphere)
Manfred Weber, Frankfurt [MW1] (A) (28)
Markus Wenke, Heidelberg [MW3] (A) (15)
Prof. Dr. David Wineland, Boulder, USA [DW] (A) (Essay Atom and Ion Traps)
Dr. Harald Wirth, Saint Genis-Pouilly, F [HW1] (A) (20) Steffen Wolf, Freiburg [SW] (A) (16)
Dr. Michael Zillgitt, Frankfurt [MZ] (A) (02)
Prof. Dr. Helmut Zimmermann, Jena [HZ] (A) (32)
Dr. Kai Zuber, Dortmund [KZ] (A) (19)

Dr. Ulrich Kilian (responsible)
Christine Weber

Priv.-Doz. Dr. Dieter Hoffmann, Berlin

The author's abbreviation is in square brackets, the number in round brackets is the subject area number, a list of subject areas can be found in the foreword.

Markus Aspelmeyer, Munich [MA1] (A) (20)
Dr. Katja Bammel, Cagliari, I [KB2] (A) (13)
Doz. Hans-Georg Bartel, Berlin [HGB] (A) (02)
Steffen Bauer, Karlsruhe [SB2] (A) (20, 22)
Dr. Günther Beikert, Viernheim [GB1] (A) (04, 10, 25)
Prof. Dr. Hans Berckhemer, Frankfurt [HB1] (A, B) (29)
Dr. Werner Biberacher, Garching [WB] (B) (20)
Prof. Tamás S. Biró, Budapest [TB2] (A) (15)
Prof. Dr. Helmut Bokemeyer, Darmstadt [HB2] (A, B) (18)
Dr. Ulf Borgeest, Hamburg [UB2] (A) (Essay Quasars)
Dr. Thomas Bührke, Leimen [TB] (A) (32)
Jochen Büttner, Berlin [JB] (A) (02)
Dr. Matthias Delbrück, Dossenheim [MD] (A) (12, 24, 29)
Karl Eberl, Stuttgart [KE] (A) (Essay Molecular Beam Epitaxy)
Dr. Dietrich Einzel, Garching [DE] (A) (20)
Dr. Wolfgang Eisenberg, Leipzig [WE] (A) (15)
Dr. Frank Eisenhaber, Vienna [FE] (A) (27)
Dr. Roger Erb, Kassel [RE1] (A) (33 essay Optical phenomena in the atmosphere)
Dr. Christian Eurich, Bremen [CE] (A) (Essay Neural Networks)
Dr. Angelika Fallert-Müller, Groß-Zimmer [AFM] (A) (16, 26)
Stephan Fichtner, Heidelberg [SF] (A) (31)
Dr. Thomas Filk, Freiburg [TF3] (A) (10, 15 essay percolation theory)
Natalie Fischer, Walldorf [NF] (A) (32)
Dr. Harald Fuchs, Münster [HF] (A) (Essay scanning probe microscopy)
Dr. Thomas Fuhrmann, Mannheim [TF1] (A) (14)
Christian Fulda, Hanover [CF] (A) (07)
Dr. Harald Genz, Darmstadt [HG1] (A) (18)
Michael Gerding, Kühlungsborn [MG2] (A) (13)
Prof. Dr. Gerd Graßhoff, Bern [GG] (A) (02)
Andrea Greiner, Heidelberg [AG1] (A) (06)
Uwe Grigoleit, Weinheim [UG] (A) (13)
Prof. Dr. Michael Grodzicki, Salzburg [MG1] (B) (01, 16)
Gunther Hadwich, Munich [GH] (A) (20)
Dr. Andreas Heilmann, Halle [AH1] (A) (20, 21)
Carsten Heinisch, Kaiserslautern [CH] (A) (03)
Dr. Christoph Heinze, Hamburg [CH3] (A) (29)
Dr. Marc Hemberger, Heidelberg [MH2] (A) (19)
Florian Herold, Munich [FH] (A) (20)
Dr. Hermann Hinsch, Heidelberg [HH2] (A) (22)
Priv.-Doz. Dr. Dieter Hoffmann, Berlin [DH2] (A, B) (02)
Dr. Georg Hoffmann, Gif-sur-Yvette, FR [GH1] (A) (29)
Dr. Gert Jacobi, Hamburg [GJ] (B) (09)
Renate Jerecic, Heidelberg [RJ] (A) (28)
Dr. Catherine Journet, Stuttgart [CJ] (A) (Essay nanotubes)
Prof. Dr. Josef Kallrath, Ludwigshafen, [JK] (A) (04 Essay Numerical Methods in Physics)
Priv.-Doz. Dr. Claus Kiefer, Freiburg [CK] (A) (14, 15 Essay Quantum Gravity)
Dr. Ulrich Kilian, Heidelberg [UK] (A) (19)
Dr. Uwe Klemradt, Munich [UK1] (A) (20, essay phase transitions and critical phenomena)
Dr. Achim Knoll, Karlsruhe [AK1] (A) (20)
Dr. Alexei Kojevnikov, College Park, USA [AK3] (A) (02)
Dr. Berndt Koslowski, Ulm [BK] (A) (Essay Surface and Interface Physics)
Dr. Bernd Krause, Munich [BK1] (A) (19)
Dr. Jens Kreisel, Grenoble [JK2] (A) (20)
Dr. Gero Kube, Mainz [GK] (A) (18)
Ralph Kühnle, Heidelberg [RK1] (A) (05)
Volker Lauff, Magdeburg [VL] (A) (04)
Priv.-Doz. Dr. Axel Lorke, Munich [AL] (A) (20)
Dr. Andreas Markwitz, Lower Hutt, NZ [AM1] (A) (21)
Holger Mathiszik, Celle [HM3] (A) (29)
Dr. Dirk Metzger, Mannheim [DM] (A) (07)
Prof. Dr. Karl von Meyenn, Munich [KVM] (A) (02)
Dr. Rudi Michalak, Augsburg [RM1] (A) (23)
Helmut Milde, Dresden [HM1] (A) (09)
Günter Milde, Dresden [GM1] (A) (12)
Marita Milde, Dresden [MM2] (A) (12)
Dr. Andreas Müller, Kiel [AM2] (A) (33)
Dr. Nikolaus Nestle, Leipzig [NN] (A, B) (05, 20 essays molecular beam epitaxy, surface and interface physics and scanning probe microscopy)
Dr. Thomas Otto, Geneva [TO] (A) (06)
Dr. Ulrich Parlitz, Göttingen [UP1] (A) (11)
Christof Pflumm, Karlsruhe [CP] (A) (06, 08)
Dr. Oliver Probst, Monterrey, Mexico [OP] (A) (30)
Dr. Roland Andreas Puntigam, Munich [RAP] (A) (14)
Dr. Andrea Quintel, Stuttgart [AQ] (A) (Essay nanotubes)
Dr. Gunnar Radons, Mannheim [GR1] (A) (01, 02, 32)
Dr. Max Rauner, Weinheim [MR3] (A) (15 Essay Quantum Informatics)
Robert Raussendorf, Munich [RR1] (A) (19)
Ingrid Reiser, Manhattan, USA [IR] (A) (16)
Dr. Uwe Renner, Leipzig [UR] (A) (10)
Dr. Ursula Resch-Esser, Berlin [URE] (A) (21)
Dr. Peter Oliver Roll, Ingelheim [OR1] (A, B) (15 essay quantum mechanics and its interpretations)
Prof. Dr. Siegmar Roth, Stuttgart [SR] (A) (Essay Nanotubes)
Hans-Jörg Rutsch, Walldorf [HJR] (A) (29)
Dr. Margit Sarstedt, Leuven, B [MS2] (A) (25)
Rolf Sauermost, Waldkirch [RS1] (A) (02)
Matthias Schemmel, Berlin [MS4] (A) (02)
Michael Schmid, Stuttgart [MS5] (A) (Essay nanotubes)
Dr. Martin Schön, Constance [MS] (A) (14)
Jörg Schuler, Taunusstein [JS1] (A) (06, 08)
Dr. Joachim Schüller, Dossenheim [JS2] (A) (10)
Richard Schwalbach, Mainz [RS2] (A) (17)
Prof. Dr. Paul Steinhardt, Princeton, USA [PS] (A) (Essay quasicrystals and quasi-unit cells)
Prof. Dr. Klaus Stierstadt, Munich [KS] (B)
Dr. Siegmund Stintzing, Munich [SS1] (A) (22)
Cornelius Suchy, Brussels [CS2] (A) (20)
Dr. Volker Theileis, Munich [VT] (A) (20)
Prof. Dr. Gerald 't Hooft, Utrecht, NL [GT2] (A) (essay renormalization)
Dr. Annette Vogt, Berlin [AV] (A) (02)
Dr. Thomas Volkmann, Cologne [TV] (A) (20)
Rolf vom Stein, Cologne [RVS] (A) (29)
Patrick Voss-de Haan, Mainz [PVDH] (A) (17)
Dr. Thomas Wagner, Heidelberg [TW2] (A) (29)
Dr. Hildegard Wasmuth-Fries, Ludwigshafen [HWF] (A) (26)
Manfred Weber, Frankfurt [MW1] (A) (28)
Priv.-Doz. Dr. Burghard Weiss, Lübeck [BW2] (A) (02)
Prof. Dr. Klaus Winter, Berlin [KW] (A) (essay neutrino physics)
Dr. Achim Wixforth, Munich [AW1] (A) (20)
Dr. Steffen Wolf, Berkeley, USA [SW] (A) (16)
Priv.-Doz. Dr. Jochen Wosnitza, Karlsruhe [JW] (A) (23 essay organic superconductors)
Priv.-Doz. Dr. Jörg Zegenhagen, Stuttgart [JZ3] (A) (21 essay surface reconstructions)
Dr. Kai Zuber, Dortmund [KZ] (A) (19)
Dr. Werner Zwerger, Munich [WZ] (A) (20)

Dr. Ulrich Kilian (responsible)
Christine Weber

Priv.-Doz. Dr. Dieter Hoffmann, Berlin

The author's abbreviation is in square brackets, the number in round brackets is the subject area number, a list of subject areas can be found in the foreword.

Prof. Dr. Klaus Andres, Garching [KA] (A) (10)
Markus Aspelmeyer, Munich [MA1] (A) (20)
Dr. Katja Bammel, Cagliari, I [KB2] (A) (13)
Doz. Hans-Georg Bartel, Berlin [HGB] (A) (02)
Steffen Bauer, Karlsruhe [SB2] (A) (20, 22)
Dr. Günther Beikert, Viernheim [GB1] (A) (04, 10, 25)
Prof. Dr. Hans Berckhemer, Frankfurt [HB1] (A, B) (29 Essay Seismology)
Dr. Werner Biberacher, Garching [WB] (B) (20)
Prof. Tamás S. Biró, Budapest [TB2] (A) (15)
Prof. Dr. Helmut Bokemeyer, Darmstadt [HB2] (A, B) (18)
Dr. Thomas Bührke, Leimen [TB] (A) (32)
Jochen Büttner, Berlin [JB] (A) (02)
Dr. Matthias Delbrück, Dossenheim [MD] (A) (12, 24, 29)
Prof. Dr. Martin Dressel, Stuttgart (A) (essay spin density waves)
Dr. Michael Eckert, Munich [ME] (A) (02)
Dr. Dietrich Einzel, Garching (A) (essay superconductivity and superfluidity)
Dr. Wolfgang Eisenberg, Leipzig [WE] (A) (15)
Dr. Frank Eisenhaber, Vienna [FE] (A) (27)
Dr. Roger Erb, Kassel [RE1] (A) (33)
Dr. Angelika Fallert-Müller, Groß-Zimmer [AFM] (A) (16, 26)
Stephan Fichtner, Heidelberg [SF] (A) (31)
Dr. Thomas Filk, Freiburg [TF3] (A) (10, 15)
Natalie Fischer, Walldorf [NF] (A) (32)
Dr. Thomas Fuhrmann, Mannheim [TF1] (A) (14)
Christian Fulda, Hanover [CF] (A) (07)
Frank Gabler, Frankfurt [FG1] (A) (22)
Dr. Harald Genz, Darmstadt [HG1] (A) (18)
Prof. Dr. Henning Genz, Karlsruhe [HG2] (A) (Essays Symmetry and Vacuum)
Dr. Michael Gerding, Potsdam [MG2] (A) (13)
Andrea Greiner, Heidelberg [AG1] (A) (06)
Uwe Grigoleit, Weinheim [UG] (A) (13)
Gunther Hadwich, Munich [GH] (A) (20)
Dr. Andreas Heilmann, Halle [AH1] (A) (20, 21)
Carsten Heinisch, Kaiserslautern [CH] (A) (03)
Dr. Marc Hemberger, Heidelberg [MH2] (A) (19)
Dr. Sascha Hilgenfeldt, Cambridge, USA (A) (essay sonoluminescence)
Dr. Hermann Hinsch, Heidelberg [HH2] (A) (22)
Priv.-Doz. Dr. Dieter Hoffmann, Berlin [DH2] (A, B) (02)
Dr. Gert Jacobi, Hamburg [GJ] (B) (09)
Renate Jerecic, Heidelberg [RJ] (A) (28)
Prof. Dr. Josef Kallrath, Ludwigshafen [JK] (A) (04)
Priv.-Doz. Dr. Claus Kiefer, Freiburg [CK] (A) (14, 15)
Dr. Ulrich Kilian, Heidelberg [UK] (A) (19)
Thomas Kluge, Jülich [TK] (A) (20)
Dr. Achim Knoll, Karlsruhe [AK1] (A) (20)
Dr. Alexei Kojevnikov, College Park, USA [AK3] (A) (02)
Dr. Bernd Krause, Munich [BK1] (A) (19)
Dr. Gero Kube, Mainz [GK] (A) (18)
Ralph Kühnle, Heidelberg [RK1] (A) (05)
Volker Lauff, Magdeburg [VL] (A) (04)
Dr. Anton Lerf, Garching [AL1] (A) (23)
Dr. Detlef Lohse, Twente, NL (A) (essay sonoluminescence)
Priv.-Doz. Dr. Axel Lorke, Munich [AL] (A) (20)
Prof. Dr. Jan Louis, Halle (A) (essay string theory)
Dr. Andreas Markwitz, Lower Hutt, NZ [AM1] (A) (21)
Holger Mathiszik, Celle [HM3] (A) (29)
Dr. Dirk Metzger, Mannheim [DM] (A) (07)
Dr. Rudi Michalak, Dresden [RM1] (A) (23 essay low temperature physics)
Günter Milde, Dresden [GM1] (A) (12)
Helmut Milde, Dresden [HM1] (A) (09)
Marita Milde, Dresden [MM2] (A) (12)
Prof. Dr. Andreas Müller, Trier [AM2] (A) (33)
Prof. Dr. Karl Otto Münnich, Heidelberg (A) (Essay Environmental Physics)
Dr. Nikolaus Nestle, Leipzig [NN] (A, B) (05, 20)
Dr. Thomas Otto, Geneva [TO] (A) (06)
Priv.-Doz. Dr. Ulrich Parlitz, Göttingen [UP1] (A) (11)
Christof Pflumm, Karlsruhe [CP] (A) (06, 08)
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Prof. Dr. Stefan Theisen, Munich (A) (essay string theory)
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### Articles on the topic

take something that interests you! Then you have fun with it too. If you are into the natural sciences, the possibilities are endless!

Are you interested in physics? Then do something in the field of natural phenomena or in the field of technology. If you can get excited about something like that, you can also go into real astronomy / starology.

If you're interested in plants and biology, do something about it. If you are enthusiastic about chemistry / biochemistry, go into medicine or pharmacy. There are so many possibilities. You can also write about cell activity or whatever I know. But you have to choose your topic yourself!

## Complex functions - chemistry and physics

Currently - comprehensive - competent

Compact
EAN: 9783817474738 (ISBN: 3-8174-7473-3)
448 pages, paperback, 11 x 18cm, 2006

EUR 4.95
all information without guarantee

The current and comprehensive reference work for a quick overview of all fields of knowledge in mathematics, physics and inorganic chemistry.

All important formulas and laws for algebra, geometry and stochastics, mechanics, acoustics and optics, stoichiometry, redox reactions and electrochemistry as well as other important subject areas.

Explained in a way that is easy to understand with catchy and apt examples. Indispensable for school, studies, work and leisure.

For some students, math, physics and chemistry offer a vast system of knowledge and rules. It is important to get a clear overview. This collection of formulas for the subjects of mathematics, physics and organic chemistry brings order to the system and offers well-founded and appealing help. In addition to the formulas and laws of the most important subject areas, the many examples are particularly helpful, as they clearly illustrate the implementation.

Arthur Thömmes, lehrerbibliothek.de

mathematics
I. Characters, types of invoices, rules
1. General signs
2. Set theory symbols
3. Symbols of logic
4. Types of invoices
5. Sign rules
6. Calculation rules for fractions
7. The number system
II. Arithmetic and Algebra
1. Arithmetic in real numbers
2. Imaginary and complex numbers
3. The logarithmic calculation
4. Percentage calculation and interest calculation
5. Sequences and ranks
6. Determinants
7. Matrices
8. Equations and inequalities
9. Features
10. Zeros of polynomials
11. Vector calculation
III. geometry
1. Planimetry
2. Stereometry
3. Trigonometry
4. Analytical geometry in the plane
5. Analytical geometry in space
6. Figure in the plane
IV. Calculus
1. Differential calculus
2. Integral calculus
3. Curve discussion
V. Stochastics
1. Probability Theory
2. Descriptive statistics
VI. Special types of invoices
1. The calculation of errors
2. The equalization calculation
3. The approximate calculation
4. Mathematical optimization

physics
I. Mechanics
1. Physical quantities
2. Mechanics of solids
3. Mechanics of liquid and gaseous bodies
II. Acoustics
1. Vibrations and waves
2. The sound strength
3. The volume
III. optics
1. The speed of light
2. Reflection of the light
3. The mirror
4. Refraction of light
5. Optical lenses
6. Magnifications by optical devices
7. Lighting parameters
IV. Thermal theory
1. Important terms
2. Expansion of bodies when there is a change in temperature
3. Gas laws
4. The heat energy
5. Changes of state of ideal gases
6. Changes in the physical state
7. Kinetic heat theory
V. Electricity
1. Magnetism
2. Electricity
VI. Atomic theory
1. Structure of the atom

chemistry
I. Introductory basic terms
1. The substance understood
2. The mix
3. The basic substance
II. States of substances
1. The types of states of the substances
2. The changes in the state of the substances
III. Basic reactions in chemistry
1. The chemical reaction
2. The decomposition (analysis)
3. The union (synthesis)
4. Realizations
IV. Basic building blocks of chemistry
1. The elements
2. The atoms
3. The molecules
V. Mass ratios in chemical reactions
1. The law of the conservation of mass
2. The law of constant proportions
3. The law of multiple proportions
VI. Volume ratios in chemical reactions
1. The gas laws
2. The general gas equation
3. The Law of Gay-Lussac and Humboldt
VII. The formula language of chemistry
1. The meaning of the element symbols
2. The chemical formula
3. The chemical equation
VIII. Stoichiometry
1. Relative atomic mass (m ^
2. Relative molecular mass (m ^)
3. The mole term
4. Stoichiometric valence
5. Statements of a chemical equation
IX. Energy ratios in chemical reactions
1. The enthalpy (H)
2. The activation energy (AE)
3. Stability conditions
4. The catalyst
5. The entropy (S)
X. The chemical bond
1. The ion bond
2. The atomic bond
3. The polar bond
4. The metal bond
XI. Acid-base reactions
1. Acids (according to Arrhenius)
2. Bases (after Arrhenius)
3. Salts
4. Brönsted acid-base reaction
5. The neutralization
XII. The redox reaction
1. The oxidation
2. The reduction
3. The oxidation number
4. Rules for creating redox reactions
XIII. The chemical equilibrium
1. The reaction rate (v)
2. The law of mass action (MWG)
3. The solubility product (L)
4. The acid and base strength
5. The pH
XIV. Electrochemistry
1. Galvanic elements
2. The voltage series of the elements
3. The Nernst equation
4. Batteries
5. Corrosion processes
XV. The periodic table of the elements
1. The main groups
2. The periods
3. The ordinal number
XVI. The main group elements
1. The hydrogen t H
2. Main group I - alkali metals
3. The II. Main group - alkaline earth metals
4. The III. Main group - earth metals
5. The IV main group - carbon group
6. The V main group - nitrogen group
7. The VI. Main group - chalcogens
8. The VII. Main group - halogens
9. Main group VIII - noble gases
XVII. The subgroup elements
1. The la - copper group
2. The Ha-zinc group
3. The purple - scandium group
4. The IVa - titanium group
5. The Va - vanadium group
6. The Via - Chromium Group
7. The Vlla - Manganese Group
8. The Villa - Iron-Cobalt-Nickel Group
XVIII. Pollutants and the environment
1. Pollutants
2. Air pollution and air pollution control
3. Water pollution and water protection
attachment
1. Mathematics
2. Physics
3. Chemistry
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## Function

• Film former: Many polyquaternium representatives form on surfaces, such as & # 160B. Hair or fingernails, a cohesive film. The film should u. & # 160a. have a protective and stabilizing effect.
• Antistatic agents: They reduce the electrostatic charge caused by friction effects, e.g. & # 160B. by combing your hair.
• Combability aids.

Some representatives of the Polyquaternium group have additional functions. For example, & # 160B. Polyquaternium-2 make the skin supple, Polyquaternium-45, -46 and -47 as hair fixatives make it easier to shape hairstyles.

## Catalog

##### APA Citation

Landolt, H. & Bornstein, R. (1950). Numerical values ​​and functions from physics, chemistry, astronomy, geophysics and technology. Berlin: Springer

##### MLA Citation

Landolt, H. and Bornstein, R. Numerical values ​​and functions from physics, chemistry, astronomy, geophysics and technology in cooperation with J. Bartels [et al.] And with the preparatory assistance of J. D'Ans [et al.] Ed. by Arnold Eucken Springer Berlin 1950

##### Australian / Harvard Citation

Landolt, H. & Bornstein, R. 1950, Numerical values ​​and functions from physics, chemistry, astronomy, geophysics and technology in collaboration with J. Bartels [et al.] And with the preparatory assistance of J. D'Ans [et al.] Ed. by Arnold Eucken Springer Berlin