Αποτελέσματα Αναζήτησης
30 Απρ 2024 · What is a complex logarithm. Learn how to solve complex logarithmic equations with rules and examples.
Dive into the fascinating world of engineering maths as you explore the concept of complex logarithm. Grasp the core definition and meaning, broaden your understanding with relevant examples, and discover the fundamental properties of this crucial mathematical element.
In 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 =.
In this topic, we have discussed the logarithm of complex numbers, their definition, showed that log e z is a many-valued f n, discussed the General and principal value of log e z, and showed how it works. Moreover, mentioned some important properties of the logarithm and rules of the logarithm.
Example 1: Calculate log z for z = − 1 − 3 i. Solution: If z = − 1 − 3 i, then r = 2 and Θ = − 2 π 3. Hence. log (− 1 − 3 i) = ln 2 + i (− 2 π 3 + 2 n π) = ln 2 + 2 (n − 1 3) π i. with n ∈ Z. The principal value of log z is the value obtained from equation (2) when n = 0 and is denoted by Log z. Thus Log z = ln r + i Θ.
Definition: Complex Log Function. The function \(\text{log} (z)\) is defined as \[\text{log} (z) = \text{log} (|z|) + i \text{arg} (z), \nonumber \] where \(\text{log} (|z|)\) is the usual natural logarithm of a positive real number.
In these notes, we examine the logarithm, exponential and power functions, where the arguments∗ of these functions can be complex numbers. In particular, we are interested in how their properties differ from the properties of the corresponding real-valued functions.† 1. Review of the properties of the argument of a complex number
