Technology & Engineering
Complex Logarithm
A complex logarithm is a logarithm that is defined for complex numbers. It is the inverse function of the exponential function and is used to solve complex equations involving exponents. The complex logarithm has many applications in fields such as physics, engineering, and mathematics.
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4 Key excerpts on "Complex Logarithm"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Complex Logarithm and Logarithmic Scale Complex Logarithm A single branch of the Complex Logarithm. The hue of the color is used to show the arg (polar coordinate angle) of the Complex Logarithm. The saturation (intensity) of the color is used to show the modulus of the Complex Logarithm. The page with the large version of this picture has an image that shows the encoding of colors as a function of their complex values. ________________________ WORLD TECHNOLOGIES ________________________ In complex analysis, a Complex Logarithm function is an inverse of the complex expon-ential function, just as the natural logarithm ln x is the inverse of the real exponential function e x . So a logarithm of z is a complex number w such that e w = z . The notation for such a w is log z . But because every nonzero complex number z has infinitely many logarithms, care is required to give this notation an unambiguous meaning. If z = re iθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2 πi gives all the others. Problems with inverting the complex exponential function For a function to have an inverse, it must map distinct values to distinct values. But the complex exponential function does not have this property: e w +2 πi = e w for any w , since adding iθ to w has the effect of rotating e w counterclockwise θ radians. Even worse, the infinitely many numbers forming a sequence of equally spaced points along a vertical line, are all mapped to the same number by the exponential function. So the exponential function does not have an inverse function in the standard sense.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Complex Logarithm and Logarithmic Scale Complex Logarithm A single branch of the Complex Logarithm. The hue of the color is used to show the arg (polar coordinate angle) of the Complex Logarithm. The saturation (intensity) of the color is used to show the modulus of the Complex Logarithm. The page with the large version of this picture has an image that shows the encoding of colors as a function of their complex values. ________________________ WORLD TECHNOLOGIES ________________________ In complex analysis, a Complex Logarithm function is an inverse of the complex expo-nential function, just as the natural logarithm ln x is the inverse of the real exponential function e x . So a logarithm of z is a complex number w such that e w = z . The notation for such a w is log z . But because every nonzero complex number z has infinitely many logarithms, care is required to give this notation an unambiguous meaning. If z = re iθ with r > 0 (polar form), then w = ln r + iθ is one logarithm of z; adding integer multiples of 2 πi gives all the others. Problems with inverting the complex exponential function For a function to have an inverse, it must map distinct values to distinct values. But the complex exponential function does not have this property: e w +2 πi = e w for any w , since adding iθ to w has the effect of rotating e w counterclockwise θ radians. Even worse, the infinitely many numbers forming a sequence of equally spaced points along a vertical line, are all mapped to the same number by the exponential function. So the exponential function does not have an inverse function in the standard sense.
- Gautam Bandyopadhyay(Author)
- 2023(Publication Date)
- CRC Press(Publisher)
Main impetus in this regard came from astronomy where it was frequently necessary to multiply and divide large numbers. However, logarithm can be perceived from many other angles. It can be viewed as the area under the rectangular hyperbola y = 1 x in geometry. It can be used as the inverse of exponential function e x or a x. As such we may treat it as the inverse of continuous compounding problem when we are interested to know in how many years Rs. 1/- will have a matured value e x or a x. In analysis we find that it is the limit of the product of two factors which are functions of n when n tends to infinity. It can also be expressed as an infinite series. It is one of the core functions in mathematics extended to negative and complex numbers. It plays vital roles in many branches of mathematics. Mathematical expressions for inductance and capacitance of a transmission line contain logarithmic terms. Logarithm forms the basis of Richter scale and measure of pH. It has wide applications in many other fields as well. 3.2 Logarithm as artificial numbers facilitating computation “Logarithms are a set of artificial numbers invented and formed into tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers in such a manner that by their aid Addition supplies the place of Multiplication, Subtraction to that of Division, Multiplication that of Involution, and Division that of Evolution or the Extraction of Roots”. Excerpt from A Manual of Logarithms and Practical Mathematics for the use of students, Engineers, Navigators and Surveyors — by James Trotter of Edinburgh Published by Oliver & Boyd, Tweeddale Court and Simpkin, Marshall, & Co. London in 1841. In eleventh century Ibon Jonuis, an Arab mathematician proposed a method of multiplication which can save computational labour significantly. The method is known as Prosthaphaeresis. The Greek word prosthesis means addition and aphaeresis means subtraction
eBook - ePub- Pramod Kumar Meher, Thanos Stouraitis, Pramod Kumar Meher, Thanos Stouraitis(Authors)
- 2017(Publication Date)
- Wiley-IEEE Press(Publisher)
Coleman has addressed the problem using a different type of co-transformation that maps the original problem to the computation of the same function in a region far from the singularity [15]. In this procedure, the subtraction function is required to be computed more than once.7.4 Forward and Inverse Conversion
The conversion from the linear to logarithmic domain and vice versa is necessarily the computation of a logarithm and an exponential, or antilogarithm, respectively. Several authors have disclosed efficient circuits for such computations, optimized for a variety of implementation platforms. The main constraint driving the implementation is the required accuracy. Low-precision applications may find very simple approaches sufficient [18], for example,(7.43)Further improvements in the accuracy of Mitchell's approximation have been introduced by Mahalingam and Ranganathan [19]. However, high-precision applications require more elaborated processing, involving interpolations [20, 21].7.5 Complex Arithmetic in LNS
Complex numbers are conceived as points in the complex plane, represented as pairs (x, y) of reals. There exists a so-called rectangular representation of a complex number z, that is,(7.44)where x and y are real and j denotes a root of x2 + 1 = 0. Swartzlander et al. have used the rectangular representation of complex numbers in an LNS implementation of an FFT processor [5]. The polar representation of a complex number is a product,(7.45)where and(7.46)The polar representation (Eq. (7.45) ) can be combined with the LNS, forming the complex LNS (CLNS) representation [14]:(7.47)When the logarithm base b = e is used, the representation is reduced to z CLNS = log r + jθ. The adoption of the combined logarithmic-polar representation extends the simplicity of the real logarithmic multiplication and division to the corresponding complex operations, avoiding the need for the computation of cross-product terms required, for example, in the case of complex multiplication using the rectangular representation.Let two CLNS numbers be expressed as X = XL+ jXθand Y = YL+ jYθ. The CLNS expression of the product Z = ZL+ jZθ
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