Complex logarithm - Wikipedia
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 =.
Complex Logarithm - Equations, Properties, and Examples - Math …
2024年4月30日 · 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)
复对数 - 维基百科，自由的百科全书
复对数（英语： Complex logarithm ）为自然对数延伸到非零复数的函数，是以下两个定义中的一个，这两个定义彼此也密切相关： 非零复数 z {\displaystyle z} 的复对数，定义为可以使 e w = z {\displaystyle e^{w}=z} 的任意复数 w {\displaystyle w} [ 1 ] [ 2 ] 。
1.11: The Function log (z) - Mathematics LibreTexts
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.
The Logarithmic Function - Complex Analysis
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.
2.2 Complex Logarithms - Ximera
We define the multivalued complex logarithm and discuss its branches and properties. We also define complex exponential functions.
§4 complex exponentials, logarithms and powers - 知乎 - 知乎专栏
b^y=x, ie. logarithm is the inverse of exponential, y is the logarithm to x; if the base is e, then if we have e^w=z, then w is the logarithm to z; introduce: # of complex logarithm for e^w=z is infinite for z\in\mathbb{C} since there is infinite many arguments unless there is specified branch for the argument; we want to find the set of ...
Complex logarithms To solve ew = z , for z 6= 0: Let w = u +iv ;so ew = eueiv;and write z = jzjei : eueiv = jzjei , eu = jzj; eiv = ei : Infinitely many solutions w : u = logjzj; v = +2ˇk Describe all solutions by: log(z) = logjzj+i arg(z) Examples: log(2i) = log(2)+iˇ 2 +i2ˇk log( 3) = log(3)+iˇ+i2ˇk log(2 +5i) = log p 29 +i arctan(5 2 ...
complex logarithm. To better understand the branches of a multivalued function, consider a path in the complex plane that starts on the positive real axis at the point z 0 > 0. Let us choose argz 0 = 0, in which case the complex logarithm is given initially by lnz = Ln z 0. We shall evaluate how the complex logarithm changes along a
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