Lesson plan
Complex logarithm
of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. of . Its graph is thus obtained by rotating the graph of around the -axis. 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 z , defined to be any complex number w for which e^w = z . Such a number w is denoted by \log z . If z is given in polar form as z = re^{i\theta} , where r and \theta are real numbers with r>0 , then \ln r + i \theta is one logarithm of z , and all the complex logarithms of z are exactly the numbers of the form \ln r + i\left(\theta + 2\pi k\right) for integers k . These logarithms are equally spaced along a vertical line in the complex plane. * A complex-valued function \log \colon U \to \mathbb{C} , defined on some subset U of the set \mathbb{C}^* of nonzero complex numbers, satisfying e^{\log z} = z for all z in U . Such complex logarithm functions are analogous to the real logarithm function \ln \colon \mathbb{R}_{>0} \to \mathbb{R} , which is the inverse of the real exponential function and hence satisfies for all positive real numbers .…
Article summary
of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. of . Its graph is thus obtained by rotating the graph of around the -axis. 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 z , defined to be any complex number w for which e^w = z . Such a number w is denoted by \log z . If z is given in polar form as z = re^{i\theta} , where r and \theta are real numbers with r>0 , then \ln r + i \theta is one logarithm of z , and all the complex logarithms of z are exactly the numbers of the form \ln r + i\left(\theta + 2\pi k\right) for integers k . These logarithms are equally spaced along a vertical line in the complex plane. * A complex-valued function \log \colon U \to \mathbb{C} , defined on some subset U of the set \mathbb{C}^* of nonzero complex numbers, satisfying e^{\log z} = z for all z in U . Such complex logarithm functions are analogous to the real logarithm function \ln \colon \mathbb{R}_{>0} \to \mathbb{R} , which is the inverse of the real exponential function and hence satisfies for all positive real numbers .…
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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