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From complex number analysis to multivalued functions: How do mathematicians unravel the mystery?
In the world of mathematics, "multivalued functions" always seem to be hidden in dark corners, but they have a profound impact on complex number analysis and other branches of mathematics. This function, in some cases, has two or more values, which is mysterious and fascinating to many mathematicians. Through in-depth research on multi-valued functions, mathematicians have not only revealed the computational mysteries behind them, but also provided new perspectives and explanations for many theories.
"The concept of multi-valued functions cannot be interpreted from a single perspective."
Multivalued functions are generally defined as functions that have multiple values within a range of certain points. This means that somewhere in its domain, the function returns multiple possible results. In the mathematical world, this function is often confused with a set-valued function, but in fact, there is a subtle difference between the two. "From a geometric point of view, the image of a multi-valued function must be a zero-area line with no overlap." In the early days of mathematics, multivalued functions often originated from analytic continuations in complex number analysis. In a certain area, mathematicians may have mastered the value of a certain complex analysis function. When extending its domain to a larger range, the value of the function may depend on the path passed. This situation reflects a peculiar fact: not only does each path have its own specific solution, but there is no way to show which is the "more natural" result.
Take the square root function as an example. When we look for the square root of -1, the result depends on the choice of path on the complex plane: whether along the upper half plane or the lower half plane, both will eventually produce relative values. — In addition, when we consider the inverse function of a function, what we actually get is a multi-valued function. For example, the complex logarithmic function "When we study multi-valued functions, we often face a complex mathematical structure rather than a simple mapping." In the context of complex variables, multi-valued functions also have the concept of branch points. This structure not only attracts the attention of mathematicians, but also begins to enter the field of physics, providing a basis for describing problems such as particle physics and crystal defects. Certain models in physics, whether it is the vortex of a superfluid or the plastic deformation of a material, can be deeply analyzed and understood using these higher-order mathematical concepts. While exploring the wide range of applications of multivalued functions, mathematicians have discovered that the properties of such functions are often reminiscent of the behavior of periodic functions. For some functions, such as trigonometric functions, when we try to find their inverse functions, we naturally face the reality of multiple solutions. For example, when we consider the multiple possible values returned by Although the foundation of mathematics is complete and rigorous, whether the mystery of multi-valued functions can be fully explained remains an ongoing challenge. Is there a deep mathematical structure that can simplify and unify all multi-valued mappings? This is not only an issue worth exploring in mathematics, but may also affect the research direction of other disciplines such as physics. As we learn more about these mysterious multi-valued functions, will we find that they are inextricably linked to some seemingly simple phenomena in our lives?
f(x) can represent all possible corresponding values of
±i. This phenomenon also exists in many other functions, such as nth roots, logarithms, and inverse trigonometric functions. Its complexity fascinates mathematicians and promotes the development of related theories. log(z) is the multivalued inverse function of the exponential function ez, which involves many solutions for each w , which makes it impossible to fully describe its behavior with a single value.
tan(π/4), how to select relevant single values in different ranges also poses a challenge for mathematicians to think about.
