Arguments of exponential and trig functions

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SUMMARY

The discussion centers on the arguments of exponential functions and trigonometric functions, specifically whether these arguments can be vectors or must be scalars. It is established that while exponential functions can be extended to square matrices, they do not generally apply to vectors due to the lack of a universal multiplication function yielding vector results. Furthermore, when dealing with physical quantities, arguments are typically dimensionless, although artificial constructs can be created, such as exponentiating force in the context of F=ma.

PREREQUISITES
  • Understanding of exponential functions and their properties
  • Familiarity with trigonometric functions and their applications
  • Basic knowledge of linear algebra, particularly square matrices
  • Concept of dimensional analysis in physics
NEXT STEPS
  • Explore the properties of power series and their applications in various algebraic structures
  • Study the implications of matrix exponentiation in linear algebra
  • Investigate the dimensional analysis of physical equations and their implications
  • Learn about the limitations of vector operations in higher dimensions
USEFUL FOR

Mathematicians, physicists, and students of linear algebra who are interested in the applications of exponential and trigonometric functions in various contexts.

dyn
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What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?
 
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dyn said:
What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?
Any function that can be expressed as a power series can be extended to apply to any algebra - that is, to any ring over a scalar field.
Thus if M is a square matrix then eM can be given a meaning.
This doesn't work for vectors in general because there is no general multiply function with a vector result. (In 3D, you could try to apply it to the cross product, but it becomes rather uninteresting when raising any given vector to a power returns zero.)
If dealing with physical quantities, it will usually be the case that the argument (and result) will be dimensionless. But you could construct artificial situations. E.g. from F=ma you could write eF=ema. Each side has the strange dimension of exponentiated force, and the units could be exponentiated Newtons. Doesn't strike me as useful.
 
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