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In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. [1] Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
The complex multiplier is the multiplier principle in Keynesian economics (formulated by John Maynard Keynes).The simplistic multiplier that is the reciprocal of the marginal propensity to save is a special case used for illustrative purposes only.
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a ...
In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator J (such that J 2 = −I) which defines multiplication by the imaginary unit i. Any such space, as a real space, is oriented.
The letter stands for a vector in , is a complex number, and ¯ denotes the complex conjugate of . [1] More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J {\displaystyle J} (different ...
With this definition, the principal cube root of a negative number is a complex number, and for instance will not be −2, but rather + This difficulty can also be solved by considering the cube root as a multivalued function : if we write the original complex number x in three equivalent forms, namely
A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure : = = + . This is the exponential map for the circle group.
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