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Conformal Mappings
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
The function inverse to the exponential function is said to be the logarithmic function. Since the exponential function is not 1-1, the logarithmic function is multivalued; it is denoted logz. Let us take w=logz, so that z=ew, and set w=u+ivandz=reiϕ=reiargz.
Classification Techniques
Published in Harry G. Perros, An Introduction to IoT Analytics, 2021
The information gain metric is used in ID3 and C4.5 algorithms, and it is based on the entropy. Let X be a discrete random variable, and let pi = p(X = i), i = 1, 2, . . , n, be its probability distribution. Then the entropy of X is defined as follows:HX=−∑i=1npilog2pi.Different values for the logarithmic base can be used other than 2, such as e and 10.
Chapter S1: Elementary Functions and Their Properties
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
By definition, the logarithmic function is the inverse of the exponential function. The following equivalence relation holds: y=logax⇔x=ay, $$ y = {\text{log}}_{a} x~ \Leftrightarrow ~x = a^{y} , $$
Assessment of passive drag in swimming by numerical simulation and analytical procedure
Published in Journal of Sports Sciences, 2018
Tiago M. Barbosa, Rui Ramos, António J. Silva, Daniel A. Marinho
Another factor to consider is the selection of the best correlation line to estimate the friction drag coefficient. Researchers have previously tested a set of correlation lines described as suitable for the range 105 < Re < 107 (typical number for humans and small vessels) and 107 < Re < 109 (typical of big vessels) (refer the Methods section – analytic procedures). It was expected that a correlation line within the Re reported for humans should return the best adjustment. However, the ITTC-1957 correlation line showed a better adjustment than other friction correlation lines. The same phenomenon was reported earlier for 60 young swimmers (Barbosa et al., 2015). The ICC between the friction drag component by CFD and the ITTC-1957 was 0.987. It was slightly better than other options such as the Prandtl–Schilichting, the Schultz–Grunow or the Kempf–Karman skin-friction equations. It was unclear why correlation lines modelled for higher Re showed a better adjustment. One may claim that these equations encompass the logarithmic number of the Re, whereas the vast majority of the correlation lines for lower Re are exponential equations. Indeed, an exponential function is the inverse of a logarithmic function. The latter approach may show a better approximation to what happens on a human body. In the near future, it would be interesting to report a skin-friction correlation line dedicated to human bodies.
A new perspective on teaching the natural exponential to engineering students
Published in International Journal of Mathematical Education in Science and Technology, 2022
Mukhtar Ullah, Muhammad Naveed Aman, Olaf Wolkenhauer, Jamshed Iqbal
All these properties are shared only by exponential functions. We have no choice left but to define the natural exponential function with the natural base defined by The inverse function of the natural exponential must be a logarithmic function. That motivates us to define the natural Logarithm by Note that is the length of interval (on the x-axis) in which changes by a factor of y from its initial value of unity. Moreover, is the length of interval (on the x-axis) in which changes by a factor of from its initial value of . Calculating requires to solve for x and that is difficult algebraically. The good news is that is the solution to the IVP Similarly, is the solution to the IVP Termwise anti-differentiation gives Replacing v by gives, Subtracting this from the previous equation gives Setting gives a power series of valid for all positive reals, Figure 4 illustrates the construction of from polynomials of increasing degrees in . That completes our exploration: the exponential function arises naturally as the solution to a specific IVP and the logarithmic function arises as the solution to an associated IVP. It is important to notice that the IVPs were solved by successive anti-differentiation. This is very different from standard methods of solving these IVPs which require knowledge of the two functions (exponential and logarithmic). We have turned that around and showed how the two functions arise naturally. One last comment: (53) allows to use our solution to (55) to solve equations of the form (48) in the dimensional form.