Arguments of exponential and trig functions

• dyn
In summary, exponential and trigonometric functions can be applied to any algebra, but cannot be extended to vectors due to the lack of a general multiply function. If dealing with physical quantities, the argument and result are usually dimensionless, but artificial situations can be constructed where the units are exponentiated.
dyn
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 ?

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.

dyn

1. What is the difference between exponential and trigonometric functions?

Exponential functions involve raising a constant base to a variable power, while trigonometric functions involve ratios of angles in a triangle. Exponential functions have a constant rate of change, while trigonometric functions have a periodic or cyclical pattern.

2. How are exponential and trigonometric functions used in real life?

Exponential functions are commonly used to model population growth, compound interest, and radioactive decay. Trigonometric functions are used to describe periodic phenomena such as sound waves, light waves, and electrical currents.

3. Can exponential and trigonometric functions be graphed?

Yes, both exponential and trigonometric functions can be graphed on a coordinate plane. Exponential functions will have a curved graph, while trigonometric functions will have a periodic wave-like graph.

4. Are there any special properties or rules for working with exponential and trigonometric functions?

Yes, exponential functions have a special property called the "power rule" which states that when an exponential function is raised to a power, the exponents can be multiplied. Trigonometric functions have several special properties, such as the Pythagorean identities and the unit circle, which are used to simplify and solve equations involving trigonometric functions.

5. How are exponential and trigonometric functions related to each other?

Exponential and trigonometric functions are related through complex numbers and the Euler's formula, which states that e^(ix) = cos(x) + i sin(x). This allows for the conversion between exponential and trigonometric forms and is used in various applications such as Fourier analysis and signal processing.

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