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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 =.
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.
30 Απρ 2024 · Complex Logarithm. The complex logarithm is an extension of the concept of logarithmic functions involving complex numbers (represented by log z). Mathematically, written as. log (z) = log (r ⋅ e iθ) = ln (r) + i (θ + 2nℼ) Here, z = r ⋅ e iθ = the complex number. r = |z| = the absolute value of z.
Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable z = r e i Θ is defined by the formula. (2) log z = ln r + i (Θ + 2 n π) (n ∈ Z). 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.
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.
§ Complex Logarithm Function. The real logarithm function ln x is defined as the inverse of the exponential function — y = ln x is the unique solution of the equation x = ey. This works because ex is a one-to-one function; if x1 6=x2, then ex1 6=ex2.
11 Αυγ 2021 · Therefore, the (multiple-valued) logarithmic function of a nonzero complex variable z = reiΘ is defined by the formula. logz = lnr + i(Θ + 2nπ) (n ∈ Z). Example 2.4.1. Example 1: Calculate logz for z = − 1 − √3i. Solution: If z = − 1 − √3i, then r = 2 and Θ = − 2π 3. Hence. log(− 1 − √3i) = ln2 + i(− 2π 3 + 2nπ) = ln2 + 2(n − 1 3)πi. with n ∈ Z.
