Jacobi elliptic functionsIn mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for .
Trigonometric functionsIn mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
Theta functionIn mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function.
Lemniscate elliptic functionsIn mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead), are analogous to the trigonometric functions sine and cosine.
Inverse trigonometric functionsIn mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
Hyperbolic functionsIn mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.
MouthIn animal anatomy, the mouth, also known as the oral cavity, or in Latin cavum oris, is the opening through which many animals take in food and issue vocal sounds. It is also the cavity lying at the upper end of the alimentary canal, bounded on the outside by the lips and inside by the pharynx. In tetrapods, it contains the tongue and, except for some like birds, teeth. This cavity is also known as the buccal cavity, from the Latin bucca ("cheek").
Gray's AnatomyGray's Anatomy is a reference book of human anatomy written by Henry Gray, illustrated by Henry Vandyke Carter, and first published in London in 1858. It has gone through multiple revised editions and the current edition, the 42nd (October 2020), remains a standard reference, often considered "the doctors' bible". Earlier editions were called Anatomy: Descriptive and Surgical, Anatomy of the Human Body and Gray's Anatomy: Descriptive and Applied, but the book's name is commonly shortened to, and later editions are titled, Gray's Anatomy.
Even and odd functionsIn mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if n is an even integer, and it is an odd function if n is an odd integer.
AnatomyAnatomy () is the branch of biology concerned with the study of the structure of organisms and their parts. Anatomy is a branch of natural science that deals with the structural organization of living things. It is an old science, having its beginnings in prehistoric times. Anatomy is inherently tied to developmental biology, embryology, comparative anatomy, evolutionary biology, and phylogeny, as these are the processes by which anatomy is generated, both over immediate and long-term timescales.
Bessel functionBessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number , which represents the order of the Bessel function. Although and produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of . The most important cases are when is an integer or half-integer.
Weierstrass elliptic functionIn mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
Transcendental functionIn mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Formally, an analytic function f (z) of one real or complex variable z is transcendental if it is algebraically independent of that variable.
Human mouthIn human anatomy, the mouth is the first portion of the alimentary canal that receives food and produces saliva. The oral mucosa is the mucous membrane epithelium lining the inside of the mouth. In addition to its primary role as the beginning of the digestive system, the mouth also plays a significant role in communication. While primary aspects of the voice are produced in the throat, the tongue, lips, and jaw are also needed to produce the range of sounds included in speech.
Oral microbiologyOral microbiology is the study of the microorganisms (microbiota) of the oral cavity and their interactions between oral microorganisms or with the host. The environment present in the human mouth is suited to the growth of characteristic microorganisms found there. It provides a source of water and nutrients, as well as a moderate temperature. Resident microbes of the mouth adhere to the teeth and gums to resist mechanical flushing from the mouth to stomach where acid-sensitive microbes are destroyed by hydrochloric acid.
MandibleIn anatomy, the mandible, lower jaw or jawbone is the largest, strongest and lowest bone in the human facial skeleton. It forms the lower jaw and holds the lower teeth in place. The mandible sits beneath the maxilla. It is the only movable bone of the skull (discounting the ossicles of the middle ear). It is connected to the temporal bones by the temporomandibular joints. The bone is formed in the fetus from a fusion of the left and right mandibular prominences, and the point where these sides join, the mandibular symphysis, is still visible as a faint ridge in the midline.
Pharyngeal archThe pharyngeal arches, also known as visceral arches, are structures seen in the embryonic development of vertebrates that are recognisable precursors for many structures. In fish, the arches are known as the branchial arches, or gill arches. In the human embryo, the arches are first seen during the fourth week of development. They appear as a series of outpouchings of mesoderm on both sides of the developing pharynx. The vasculature of the pharyngeal arches is known as the aortic arches.
Oral hygieneOral hygiene is the practice of keeping one's oral cavity clean and free of disease and other problems (e.g. bad breath) by regular brushing of the teeth (dental hygiene) and adopting good hygiene habits. It is important that oral hygiene be carried out on a regular basis to enable prevention of dental disease and bad breath. The most common types of dental disease are tooth decay (cavities, dental caries) and gum diseases, including gingivitis, and periodontitis.
JawThe jaw is any opposable articulated structure at the entrance of the mouth, typically used for grasping and manipulating food. The term jaws is also broadly applied to the whole of the structures constituting the vault of the mouth and serving to open and close it and is part of the body plan of humans and most animals. In arthropods, the jaws are chitinous and oppose laterally, and may consist of mandibles or chelicerae. These jaws are often composed of numerous mouthparts.
Skeletal muscleSkeletal muscles (commonly referred to as muscles) are organs of the vertebrate muscular system and typically are attached by tendons to bones of a skeleton. The muscle cells of skeletal muscles are much longer than in the other types of muscle tissue, and are often known as muscle fibers. The muscle tissue of a skeletal muscle is striated – having a striped appearance due to the arrangement of the sarcomeres. Skeletal muscles are voluntary muscles under the control of the somatic nervous system.
