Special functions

For a comprehensive reference on special functions, see M. Abramowitz and I. A. Stegun, eds.. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series, Vol 55, Washington, DC, 1964). A Digital Library of Mathematical Functions is in preparation by the National Institute of Standards and Technology. This is scheduled to be online at http //dlmf.nist.gov sometime in 2007. It will feature hypertext links and graphics. 

Following are samples of some amazing formulas involving gamma and zeta functions, two of the special functions we have already introduced. A reflection formula for gamma functions  

Chapter 12 Partial Differential Equations and Special Functions 

Euler s relation connecting the zeta function with prime numbers, which we proved in Section 1.12  

For a long time, 1 felt intimidated by these amazing formulas involving special functions. Some resembled near-miracles like pulling rabbits out of a hat. 1 could never, in a thousand years, have come up with anything as brilliant as one of these myself. I would not have been surprised to see in some derivation and then a miracle occurs between the next-to-last and last equations (see Sidney Harris cartoon)  

Case Ilia When Oj - 00, the singularity causing the bad behavior (the smallest root, Cj) is removed, and the second solution is generated using 

Case Illb When Oj is indeterminate, then the complete solution is obtained using the smallest root Cj, taking Oq and Oj as the two arbitrary constants of integration. 

Several variable coefficient ODE arise so frequently that they are given names and their solutions are widely tabulated in mathematical handbooks. One of these special functions occupies a prominent place in science and engineering, and it is the solution of Bessel s equation 

The solutions to this equation are called Bessel functions of order p that is, Jp(x), J p(x). Because of its importance in chemical engineering, we shall reserve a complete section to the study of Bessel s equation. 

We mentioned earlier in Example 3.4 the confluent hypergeometric function of Kummer, which satisfies the equation 

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In order to provide measuring of cracks depth on complicated profile areas there is a special function in the device. The function makes it possible to perform measurements by studying on a samples with defects of the measured depth. 

Small amounts of specially functionalized monomers are often copolymerized with acryUc monomers in order to modify or improve the properties of the polymer. These functional monomers can bring about improvements either directiy or by providing sites for further reaction with metal ions, cross-linkers, or other compounds and resins. Table 9 Hsts some of the more common functional monomers used in the preparation of acryUc copolymers. 

Lehedev, N. N. Special Functions and Their Applications, Dover, New York (1972). 

Special functionalities required of the power supply. These include any power-on resets and power-fail signals needed by any microcomputers in the system, remote turn-off, output voltage or current programming, power sequencing, status signals, etc. 

In addition to acting as impact modifiers a number of polymeric additives may be considered as processing aids. These have similar chemical constitutions to the impact modifiers and include ABS, MBS, chlorinated polyethylene, acrylate-methacrylate copolymers and EVA-PVC grafts. Such materials are more compatible with the PVC and are primarily included to ensure more uniform flow and hence improve surface finish. They may also increase gelation rates. In the case of the compatible MBS polymers they have the special function already mentioned of balancing the refractive indices of the continuous and disperse phases of impact-modified compound. 

Nearly all biological processes involve the specialized functions of one or more protein molecules. Proteins function to produce other proteins, control all aspects of cellular metabolism, regulate the movement of various molecular and ionic species across membranes, convert and store cellular energy, and carry out many other activities. Essentially all of the information required to initiate, conduct, and regulate each of these functions must be contained in... 

The versatility of the plastic materials permits them to be varied to perform special functions. By applying ingenuity to these amazingly adaptable materials, we can produce products that add to the worldwide survival capability of life under severe environments and improve the quality of life under normal environments. The designer s role in fitting the possibilities to the needs is one that is increasingly important. 

Special Characteristics Information that describes the functionality of the application needs to be handled in a specific manner. This information can include such properties as size, weight, geometric form, special functional elements, processing and assembly methods, and other product specific information. 

Activated carbon filters remove a wide range of organic matter by adsorption onto the carbon bed. The bed may be derived from a number of different carbon sources, and the correct selection of bed type, capacity, and porosity is a specialized function. Activated carbon may be usefully employed in organic traps, complementing the resin bed, but its capacity and organic removal rate characteristics are flow-dependent. Excessive flows may compromise the rate of adsorption of organic matter. 

Not surprisingly, only about 20 of the chemical elements found on Earth are used by living organisms (Chapters 3 and 8). Most of them are common elements. Rare elements are used, if at all, only at extremely low concentrations for specialized functions. An example of the latter is the use of molybdenum as an essential component of nitrogenase, the enzyme that catalyzes the fixation of elemental dinitrogen. Because they are composed of common elements, living organisms exert their most profound effects on the cycles of those elements. 

This integral is a special function related to the incomplete gamma function. The solution can be considered to be analytical even though the function may be unfamiliar. Figure 8.1 illustrates the behavior of Equation (8.8) as compared with CSTRs, PFRs, and laminar flow reactors with diffusion. 

This compartment constimtes two-thirds of total body water and provides the environment for the cell (1) to make, store, and utilize energy (2) to repair itself (3) to replicate and (4) to perform special functions. 

In addition to phospholipids, cell membranes contain protein molecules that carry out special functions such as transporting ions and molecules through the membrane. 

These devices have a special function vhich allows them to perform electro-organic synthesis. Typically, they contain electrode structures to generate electrons as tunable reactants . Often, these electrodes are constructed as plate-type structures, sometimes also being the construction material for the channels themselves. 

These devices have special function, namely to irradiate the liquid phase with light to induce a photoreaction or photoinduced reaction. Hence the characteristic feature is a transparent section within the reactor, often in the visible or commonly in the UV spectral region. The devices may have integrated photo energy sources or on-line analysis units. Otherwise, this is performed by external instruments. 

The reagent used for the deteetion may be speeifie to a special functional group or specific lipids or may be a nonspecific reagent that makes all hpids visible. The most commonly used reagent that is nonspecific for any lipid group is 0.1% (w/v) 2, 7 -dichlorofuorescein in 95% methanol. This is mainly useful when the plates have been developed in acidic solvents. The lipid spots or bands can be visualized as yellow spots or bands under UV light. After the plates are developed in alkaline solvents, an aqueous solution of Rhodamine 6G (0.01%) can be used, and lipid spots can be seen as pink spots under UV. Because both these methods are nondestructive, they can be effectively used in PTLC so that the separated sample bands can be scraped off and used for further analysis. 

Special functional groups Primary, secondary, and tertiary amines, carboxylic acids, alcohols... 

The automations are often controlled by scripts that define the protocols, the methods, the timing of an automation and the interaction of the methods and automation or geometry settings. The easy way to expand scripts with user functionality enables the research user to implement special functionality to support nonstandard hardware and create complex laboratory-built automation scenarios. 

Chemokines, a superfamily of small (8 to 14 kd) chemoattractant cytokines, were originally identified by their capacity to mediate gradient-dependent cell migration, accompanied by Ca2+ flux, in vitro. Sequence homology of mammal, bird, and fish chemokines indicates that this family of proteins is highly conserved throughout evolution. Chemokines are multifunctional molecules with a vast repertoire of specialized functions in different organs. Broadly, chemokine actions are associated with cell adhesion, cytokine secretion, cellular... 

Alternatively, substrate control of diastereoselectivity can rely on attractive catalyst substrate interactions. This requires in general special functional groups which allow for a directed hydroformylation, which is summarized in Sect. 6 (vide infra). 

The Hermite polynomials introduced above represent an example of special functions which arise as solutions to various second-order differential equations. After a summary of some of the properties of these polynomials, a brief description of a few others will be presented. The choice is based on their importance in certain problems in physics and chemistry. 

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