Complex logarithmIn mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number , defined to be any complex number for which . Such a number is denoted by . If is given in polar form as , where and are real numbers with , then is one logarithm of , and all the complex logarithms of are exactly the numbers of the form for integers .
Fourier transformIn physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.
Space manufacturingSpace manufacturing is the production of tangible goods beyond Earth. Since most production capabilities are limited to low Earth orbit, the term in-orbit manufacturing is also frequently used. There are several rationales supporting in-space manufacturing: The space environment, in particular the effects of microgravity and vacuum, enable the research of and production of goods that could otherwise not be manufactured on Earth.
Adiabatic invariantA property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an adiabatic invariant. By this it is meant that if a system is varied between two end points, as the time for the variation between the end points is increased to infinity, the variation of an adiabatic invariant between the two end points goes to zero. In thermodynamics, an adiabatic process is a change that occurs without heat flow; it may be slow or fast.
Commercial use of spaceCommercial use of space is the provision of goods or services of commercial value by using equipment sent into Earth orbit or outer space. This phenomenon – aka Space Economy (or New Space Economy) – is accelerating cross-sector innovation processes combining the most advanced space and digital technologies to develop a broad portfolio of space-based services.
Contour integrationIn the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and application of the residue theorem.
Upper half-planeIn mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is defined similarly, by requiring that be negative instead. Each is an example of two-dimensional half-space. The affine transformations of the upper half-plane include shifts , , and dilations , . Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes to . Proof: First shift the center of to . Then take and dilate.
Albert Einstein'Albert Einstein' (ˈaɪnstaɪn ; ˈalbɛʁt ˈʔaɪnʃtaɪn; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely held to be one of the greatest and most influential scientists of all time. Best known for developing the theory of relativity, he also made important contributions to quantum mechanics, and was thus a central figure in the revolutionary reshaping of the scientific understanding of nature that modern physics accomplished in the first decades of the twentieth century.
Space habitatA space habitat (also called a space settlement, space colony, spacestead, space city, orbital habitat, orbital settlement, orbital colony, orbital stead or orbital city) is a more advanced form of living quarters than a space station or habitation module, in that it is intended as a permanent settlement or green habitat rather than as a simple way-station or other specialized facility. No space habitat has been constructed yet, but many design concepts, with varying degrees of realism, have come both from engineers and from science-fiction authors.
Functional integrationFunctional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields. In an ordinary integral (in the sense of Lebesgue integration) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration).
