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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.
Logarithms and exponential functions
Published in John Bird, Science and Mathematics for Engineering, 2019
An exponential function is one which contains ex, e being a constant called the exponent and having an approximate value of 2.7183. The exponent arises from the natural laws of growth and decay and is used as a base for natural or Napierian logarithms. The most common method of evaluating an exponential function is by using a scientific notation calculator.Use your calculator to check the following values: e1 = 2.7182818, correct to 8 significant figures,e−1.618 = 0.1982949, each correct to 7 significant figures,e0.12 = 1.1275, correct to 5 significant figures, e−2.785 = 0.0617291, correct to 7 decimal places.
Mathematical Background
Published in P.N. Paraskevopoulos, Modern Control Engineering, 2017
All functions presented in this section can be expressed in terms of exponential functions or derived from the exponential function, a fact which makes the exponential function very interesting. This can easily be shown as follows: (a) the sinusoidal function is a linear combination of two exponential functions, e.g., sin θ = (1/2j)(ejθ−e−jθ)(b) the unit step function for Τ = 0 is equal to the exponential function when A = 1 and a = 0, i.e., u(t) = f(t) = Aeat = 1, for A = 1 and a = 0; (c) the functions δ(t − T) and r(t − T) can be derived from the unit step function u(t − T), while u(t − T) may be derived from the exponential function. Furthermore, a periodic function can be expressed as a linear combination of exponential functions (Fourier series). Moreover, it is worth mentioning that the exponential function is used to describe many physical phenomena, such as the response of a system and radiation of nuclear isotopes.
Data-Driven Forecasting of Nonlinear System with Herding via Multi-Dimensional Taylor Network
Published in Cybernetics and Systems, 2022
Hong-Sen Yan, Guo-Biao Wang, Bo Zhou, Xiao-Qin Wan, Jiao-Jun Zhang
MTN for modeling and identifying is the simple function of states and inputs convenient for analysis and solving, especially for the optimal control by the minimum principle to obtain the initial parameters of the MTN controller for securing the corresponding closed-loop system’s stability and optimal performance. MTN guarantees real-time, with only addition or multiplication operations allowable, its computing complexity similar to that of just one neuron in NN. NNs are often treated as black boxes. Even if represented by a mathematical expression based on its internal structure, the NN is still an incredibly complicated function consisting of exponential functions level by level. No matter how many times an exponential function is differentiated, it remains an exponential one.
Pavement maintenance and rehabilitation planning optimisation under budget and pavement deterioration uncertainty
Published in International Journal of Pavement Engineering, 2022
Amirhossein Fani, Amir Golroo, S. Ali Mirhassani, Amir H. Gandomi
In fact, the deterioration process is modelled as an exponential function of time. This model is widely used as the performance model in network and project level M&R optimisation (Tsunokawa and Schofer 1994, Li and Madanat 2002, Ouyang and Madanat 2004, Seyedshohadaie et al.2010).