What Does exp Mean in Mathematical Expressions?

  • #1
cryforhelp104
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1
Homework Statement
What would "exp" in a question about approximating functions with Taylor Series mean?
Relevant Equations
exp(-ax^2) about the value x = 0 to second order in x
In my introductory modern physics class, I was asked to compute the Taylor Series for exp(-ax^2) about the value x = 0 to second order in x. I am unfamiliar with the what "exp" before the function means, despite having approximated functions with Taylor Series before. I think there was some gap in my previous math class. I'd appreciate a brief explanation (please don't work the problem, just explain the "exp" part)
 
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  • #2
It means ##e^{-ax^2}##, the exponential function. With this specific argument, it is also known as a Gaussian function (a very useful function).
 
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  • #3
cryforhelp104 said:
Homework Statement: What would "exp" in a question about approximating functions with Taylor Series mean?
Relevant Equations: exp(-ax^2) about the value x = 0 to second order in x

In my introductory modern physics class, I was asked to compute the Taylor Series for exp(-ax^2) about the value x = 0 to second order in x. I am unfamiliar with the what "exp" before the function means, despite having approximated functions with Taylor Series before. I think there was some gap in my previous math class. I'd appreciate a brief explanation (please don't work the problem, just explain the "exp" part)
Frabjous said:
It means ##e^{-ax^2}##, the exponential function. With this specific argument, it is also known as a gaussian function.
Welcome to PF, @cryforhelp104 -- Do you have what you need now to actually show some effort on this schoolwork problem of yours? :wink:
 
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  • #4
Thank you! So the question is to compute the Taylor Series for (e^(-ax^2)) about the value x = 0 to second order in x?
 
  • #5
cryforhelp104 said:
Thank you! So the question is to compute the Taylor Series for (e^(-ax^2)) about the value x = 0 to second order in x?
Yes.
 
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  • #6
A Taylor series in powers of x (expanded about x = 0) is a Maclaurin series. The Maclaurin series for ##e^x## is one of the simplest infinite series, where ##e^x = 1 + \frac x 1 + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots##.
Just do a substitution to get the Maclaurin series for ##e^{-ax^2}## for as many terms as are required and you're done.
 
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FAQ: What Does exp Mean in Mathematical Expressions?

What does "exp" stand for in mathematical expressions?

The term "exp" stands for the exponential function, which is a mathematical function denoted by exp(x). It is equivalent to raising the constant e (approximately 2.71828) to the power of x, where e is the base of the natural logarithm.

How is the exp function related to the natural logarithm?

The exponential function exp(x) is the inverse of the natural logarithm function, denoted as ln(x). This means that if y = exp(x), then x = ln(y). Essentially, exp and ln undo each other.

In what contexts is the exp function commonly used?

The exp function is widely used in various fields such as calculus, differential equations, complex analysis, and in modeling exponential growth or decay in fields like biology, economics, and physics.

How do you compute exp(x) for a given x?

To compute exp(x) for a given x, you can use a calculator or software that has the exponential function built-in. Most scientific calculators have an "exp" button, and in programming languages, functions like exp(x) are available in libraries such as math in Python or cmath in C++.

What are some properties of the exp function?

The exp function has several important properties: it is always positive, it grows faster than any polynomial as x approaches infinity, and it satisfies the equation exp(a + b) = exp(a) * exp(b). Additionally, its derivative is itself, meaning d(exp(x))/dx = exp(x).

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