Function as a Solution to Specific Conditions

In summary, the conversation is about solving a statistical mechanics problem with the given equation ##\frac{f\left(0\right)}{f\left(a\right)}f\left(x+a\right)=f\left(x\right)## and the solution is an exponential function, which was determined by recognizing the pattern of addition in the argument resulting in an inverse multiplication of the function. The topic falls under "functional equations" and is commonly studied in calculus and in solving limits using Taylor series. The general solution to the equation is ##f(x) = Ce^{dx}## with arbitrary constants C and d, which can also be complex.
  • #1
tade
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I'm trying to solve some statistical mechanics. This problem appeared.

##\frac{f\left(0\right)}{f\left(a\right)}f\left(x+a\right)=f\left(x\right)##

Any idea as to which function will satisfy this equation?
 
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  • #2
##f(x) = Ce^{\pm x}##
 
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  • #3
DrClaude said:
##f(x) = Ce^{\pm x}##
Nice. How did you work that out? Is an exponential function the only solution to this problem?
 
  • #4
tade said:
Nice. How did you work that out?
From the observation that ##f(x) \propto f(x+a) / f(a)## requires that an addition in the argument becomes an (inverse) multiplication of the function, and I recognized the exponential. (I actually thought about log first, but then realized I had it backwards :smile:)

tade said:
Is an exponential function the only solution to this problem?
No idea. Apart from the answer I gave and ##f(x) = \mathrm{const.}##, I don't see any.
 
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  • #5
DrClaude said:
From the observation that ##f(x) \propto f(x+a) / f(a)## requires that an addition in the argument becomes an (inverse) multiplication of the function, and I recognized the exponential. (I actually thought about log first, but then realized I had it backwards :smile:)
That's pretty neat. What branch of mathematics is that under?
 
  • #6
tade said:
That's pretty neat. What branch of mathematics is that under?
I'm not a mathematician. It just comes from years of working with, and getting a feeling for, those kind of mathematical functions.
 
  • #7
DrClaude said:
I'm not a mathematician. It just comes from years of working with, and getting a feeling for, those kind of mathematical functions.
No, I was just wondering what topic or branch this question falls under.
 
  • #9
The general solution is ##f(x) = Ce^{d x}##, C and d arbitrary constants. C and d can be complex if you work with complex numbers.
It is a typical homework problem to show this.
 
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What is the definition of function?

A function is a mathematical relationship between two or more variables in which each input (independent variable) has a unique output (dependent variable). It is often represented as f(x) or y = f(x), where x is the input and f(x) is the output.

How does function help solve specific conditions?

Function can help solve specific conditions by providing a clear and systematic way to represent the relationship between variables. By understanding the function, we can manipulate the variables to find the desired output and solve the specific condition.

What are the types of functions commonly used to solve specific conditions?

The types of functions commonly used to solve specific conditions include linear functions, quadratic functions, exponential functions, logarithmic functions, and trigonometric functions. Each type has its own unique properties and can be used to solve specific conditions in different scenarios.

Can function be used in other fields besides mathematics?

Yes, function can be used in many other fields besides mathematics. In computer science, functions are a fundamental part of programming and are used to encapsulate and organize code. In biology, functions can represent relationships between biological variables. In economics, functions can be used to model supply and demand.

What are the limitations of using function to solve specific conditions?

Although functions are a powerful tool for solving specific conditions, they have some limitations. For example, functions can only represent a relationship between two variables, so they may not be suitable for more complex situations with multiple variables. Additionally, functions may not always accurately reflect real-world scenarios due to simplifications and assumptions made in the function's representation.

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