Considerations on Function f(s)=0^{-s} for Values of 's

  • Thread starter zetafunction
  • Start date
  • Tags
    Function
In summary, the function f(s)=0^{-s} has different behaviors for different values of Re(s), and its value for any given value of s should be considered in relation to its definition and properties.
  • #1
zetafunction
391
0
how could or should we consider the function

[tex] f(s)=0^{-s} [/tex] for values of 's' ??

if Re (s) is smaller than 0 then [tex]f(s)=0 [/tex]

but if Re (s) is bigger than 0 then [tex] f(s)= \infty [/tex]

If s=0 as a limite then [tex]f(0)=0^{0}=1 [/tex]

f(s) can be considered (plus a minus or + sign) as the Mellin transform [tex]f(s)=\int_{0}^{\infty}dxx^{s-1} [/tex]

if we imposed certain symmetry or regularization conditions so [tex] f(s)=f(1-s) [/tex] we would have the 'regularized' value 0 ,

then what value should i take for f(s) for every value of 's' ?
 
Physics news on Phys.org
  • #2


I would approach this question by first considering the definition and properties of the function f(s). It is clear that f(s) is undefined for s=0, since dividing by zero is not allowed in mathematics. However, as the forum post suggests, we can assign a value to f(0) by considering it as a limit. In this case, the limit of f(s) as s approaches 0 is 1, which can be seen by using the property that 0^0=1.

Next, we can examine the behavior of f(s) for different values of Re(s). As stated in the post, if Re(s) is smaller than 0, then f(s) is equal to 0. This can be seen from the definition of the Mellin transform, where the integral is only defined for positive values of x. On the other hand, if Re(s) is larger than 0, then f(s) becomes infinite, which can also be seen from the definition of the Mellin transform.

Based on these observations, it seems that f(s) does not have a single, well-defined value for all values of s. Instead, it can be seen as a function that takes on different values depending on the value of Re(s). This is not uncommon in mathematics, as many functions have different behaviors for different values of their variables.

In terms of how we should consider the function f(s), it may be useful to further explore its properties and applications. For example, the Mellin transform has many important applications in mathematics and physics, and understanding the behavior of f(s) can help us better understand these applications. Additionally, it may be helpful to consider the function in the context of other functions and their properties, as this can give us a better understanding of its behavior.

Overall, it is important to carefully consider the definition and properties of a function before assigning values to it. In the case of f(s), it may be more useful to think of it as a function with different behaviors for different values of s, rather than a single, well-defined value for all values of s.
 

1. What is the function f(s)=0^{-s}?

The function f(s)=0^{-s} is an exponential function with a base of 0 and an exponent of -s. This means that the function will approach 0 as s approaches infinity, and will be undefined when s = 0.

2. What are the possible values of s for the function f(s)=0^{-s}?

The possible values of s for this function are all real numbers except for s = 0. This is because 0^{-s} will result in division by 0 when s = 0.

3. How does the value of s affect the function f(s)=0^{-s}?

The value of s will determine the behavior of the function. As s becomes larger and larger, the function will approach 0. On the other hand, as s becomes more negative, the function will approach infinity.

4. What is the graph of the function f(s)=0^{-s}?

The graph of this function is a horizontal line at y = 0 for all values of s except for s = 0, where it is undefined. This means that the function will have a hole at s = 0 on the graph.

5. What are some real-world applications of the function f(s)=0^{-s}?

This function has limited real-world applications since it is undefined for s = 0 and approaches 0 for all other values of s. However, it can be used in situations where a value is approaching 0 as s increases, such as in exponential decay or in the calculation of limits.

Similar threads

Replies
2
Views
691
Replies
1
Views
833
Replies
4
Views
644
Replies
3
Views
1K
Replies
11
Views
2K
Replies
3
Views
1K
Replies
4
Views
844
Replies
1
Views
1K
Replies
2
Views
337
Back
Top