complex variable analysis, branch of mathematics that deals with the calculus of functions of a complex variable, i.e., a variable of the form z=x+iy, where x and y are real and i=-1 (see number). A function w=f(z) of a complex variable z is separable into two parts, w = g₁(x,y) + ig₂(x,y), where g₁ and g₂ are real-valued functions of the real variables x and y. The theory of functions of a complex variable is concerned mainly with functions that have a derivative at every point of a given domain of values for z; such functions are called analytic, regular, or holomorphic. If a function is analytic in a given domain, then it also has continuous derivatives of higher order and can be expanded in an infinite series in terms of these derivatives (i.e., a Taylor's series). The function can also be expressed in the infinite serieswhere z₀ is a point in the domain. Also of interest in complex variable analysis are the points in a domain, called singular points, where a function fails to have a derivative. The theory of functions of a complex variable was developed during the 19th cent. by A. L. Cauchy, C. F. Gauss, B. Riemann, K. T. Weierstrass, and others.

complex number: see number.

complex ion, charged molecular aggregate (see ion), consisting of a metallic atom or ion to which is attached one or more electron-donating molecules. In some complex ions, such as sulfate, SO₄^-2, the atoms are so tightly bound together that they act as a single unit. Many complex ions, however, such as tetrammine zinc (II), Zn(NH₃)₄⁺², are only loosely aggregated and tend to dissociate in a water solution until an equilibrium is established between the complex ion and its components (see chemical equilibrium). Such complex ions, or coordinated complexes as they are also called, generally consist of a positively charged central metal atom or ion, like the zinc in tetramine zinc, surrounded by electron-donating, or basic, groups called ligands; in the tetrammine zinc complex, the NH₃ groups are the ligands. The number of bonds connecting the ligands to the central atom or ion is its coordination number, or ligancy. Transition metals (see transition elements) are especially suited for forming complex ions because they have filled or partially filled electron orbitals that can participate in bonding the ligands to the metal. The bonding holding the ligands to the central atom or ion is similar to covalent bonding between atoms but is more complex (see chemical bond). All the ligands surrounding the central ion need not be the same, and some positions can be occupied by solvent molecules. Because ligands remain in a fixed position around a central atom or ion, in many complexes different isomers, or arrangements, of the ligand groups are possible. When there are four or more ligands around a central atom, different stereoisomers, or spatial configurations, are possible (see stereochemistry). Many complex ions are colored; the specific color of a complex depends on both the central atom or ion and the ligands. For example, when cobaltous chloride is dissolved in water, a pale pink solution, sometimes called invisible ink, results because of the presence of the hydrated cobaltous ion, Co(H₂O)₆⁺²; this solution does not show up well on paper, but if the paper is heated to drive the water off, visibility improves because of the formation of a blue tetrachlorocobalt (II)^-2 complex. Some of the more important complex ions are vitamin B₁₂, chlorophyll, and the heme component of hemoglobin, in which the central metal ions are cobalt, magnesium, and iron, respectively, and the ligands are complex organic systems. Many enzymes contain a metal ion about which parts of the protein are coordinated.

Oedipus complex, Freudian term, drawn from the myth of Oedipus, designating attraction on the part of the child toward the parent of the opposite sex and rivalry and hostility toward the parent of its own. It occurs during the phallic stage of the psycho-sexual development of the personality, approximately years three to five. Resolution of the Oedipus complex is believed to occur by identification with the parent of the same sex and by the renunciation of sexual interest in the parent of the opposite sex. Freud considered this complex the cornerstone of the superego and the nucleus of all human relationships. Many psychiatrists, while acknowledging the significance of the Oedipal relationships to personality development in our culture, ascribe love and attraction toward one parent and hatred and antagonism toward the other not necessarily to sexual rivalry but to resentment of parental authoritarian power.

Acute sense of personal inferiority, often resulting in either timidity or (through overcompensation) exaggerated aggressiveness. Though once a standard psychological concept, particularly among followers of Alfred Adler, it has lost much of its usefulness through imprecise popular misuse.

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Class of substances with chemical structures made up of a central metal atom surrounded by nonmetal atoms or groups of atoms, known as ligands. Examples of coordination compounds include hemoglobin, vitamin B₁₂, and chlorophyll, as well as dyes, pigments, and catalysts. Coordination compounds are used in hydrometallurgical processes for the extraction of metals such as nickel, cobalt, and copper from their ores and in important catalytic processes to bring about polymerization of organic compounds such as polyethylene and polypropylene. Coordination compounds are also used in the analysis of other compounds and in the sequestering of metal ions.

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In psychoanalytic theory, a desire for sexual involvement with the parent of the opposite sex and a sense of rivalry with the parent of the same sex. The term was introduced by Sigmund Freud in his Interpretation of Dreams (1899) and is derived from the mythological Oedipus, who killed his father and married his mother; its female analogue is the Electra complex. Considered a normal stage in the development of children ages three to five, it ends when the child identifies with the parent of the same sex and represses its sexual instincts. Freud believed that the process of overcoming the Oedipus complex gave rise to the superego.

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