Writing Co-efficients Elegantly w/ Real Numbers

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In summary: The gamma function is the unique extension of the factorial function to the real numbers. It is complex differentiable at all points except 0,-1,-2,...
  • #1
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Hi guys, just wondering if you can give me some advice on how to write certain co-efficients with in a compact elegant way.

The co-efficients are given by the following rule:

a0=1
a1=z
a2=z(z-1)
a3=z(z-1)(z-2)
.
.
.
an=z(z-1)(z-2)...(z-(n-1))

where a = any real number

I was thinking of using the following definition, an = z!/(z-n)! which seems to give me the correct results but I am worried about whether or not the ! can be defined on real numbers. Like is there a problem with this definition?
 
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  • #2
The ! can't really be defined on reals analogously to on N, but the expression "z!/(z-n)!" as a whole makes a lot of sense. Analogously to N, it means multiply z by z-1 by z-2, etc up to z-n+1.
 
  • #3
How about (for n>1)

[tex] a_n = \prod_{i=0}^{n-1} (z-i)[/tex]

Written differently, you can define it recursively by

[tex]a_0 = 1[/tex]

and

[tex]a_n = a_{n-1} \cdot (z - (n-1)), \ \text{for } \, n \geq 1[/tex]
 
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  • #4
morphism said:
How about (for n>1)

[tex] a_n = \prod_{i=0}^{n-1} (z-i)[/tex]

Written differently, you can define it recursively by

[tex]a_0 = 1[/tex]

and

[tex]a_n = a_{n-1} \cdot (z - (n-1)), \ \text{for } \, n \geq 1[/tex]


What's that thing called anyway? I know that it is to multiplication what the sigma is to addition, but is it just a capital pi or what?
 
  • #5
Pythagorean said:
What's that thing called anyway? I know that it is to multiplication what the sigma is to addition, but is it just a capital pi or what?

I am pretty sure it is just a capital pi.
 
  • #6
I learned it as "product". But that was fifty years ago...
 
  • #7
Lol maybe you misunderstood, yes they mean it as product as well jim, but they're just saying, the actual letter used to show that is a capital pi from the greek alphabet.

And just incase everyones forgotten, these can be very easily written with binomial notation?

EDIT: Shoot my last statement, though it could still help a tiny bit.

[tex]{r \choose k} &{}= {1 \over k!}\prod_{n=0}^{k-1}(r-n)=\frac{r(r-1)(r-2)\cdots(r-(k-1))}{k!}[/tex]
 
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  • #8
Try [tex]\frac{\Gamma(z+1)}{\Gamma(z-n+1)}[/tex]
The gamma function is the unique extension of the factorial function to the real numbers. [tex]\Gamma(z+1) = z![/tex]

By the way, if you are only doing this with integers, the accepted notation is [tex](z)_n[/tex] It is known as the falling factorial, or Pochhammer symbol
 
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  • #9
LukeD said:
The gamma function is the unique extension of the factorial function to the real numbers.
It's not unique. For example,

[tex]f(z) = \Gamma(z + 1) + \sin (\pi z)[/tex]

also agrees with the factorial function on the natural numbers. And so does

[tex]f(z) = \begin{cases}
0 & z < 0\\
\lfloor z \rfloor ! & z \geq 0
[/tex]

and

[tex]f(z) = \begin{cases}
z! & z \in \mathbb{N} \\
-14 & z \notin \mathbb{N}
[/tex]
 
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  • #10
Hurkyl said:
It's not unique. For example,

[tex]f(z) = \Gamma(z + 1) + \sin (\pi z)[/tex]

also agrees with the factorial function on the natural numbers. And so does

[tex]f(z) = \begin{cases}
0 & z < 0\\
\lfloor z \rfloor ! & z \geq 0
[/tex]

and

[tex]f(z) = \begin{cases}
z! & z \in \mathbb{N} \\
-14 & z \notin \mathbb{N}
[/tex]
What I meant is that it's unique in that it extends all of the properties of the factorial function to the real (and complex) numbers so that for all complex numbers (except 0) [tex]\Gamma(z+1) = z*\Gamma(z)[/tex] and so that it is complex differentiable at all points except 0,-1,-2,...

However, I haven't studied the Gamma function, so I don't know much else about it.
 
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1. What are coefficients and why are they important in writing with real numbers?

Coefficients are numbers that are attached to variables in mathematical expressions. They are important because they tell us how many times a variable is being multiplied by itself. This is crucial in writing with real numbers because it helps us better understand the relationships and patterns between the numbers.

2. Can you provide an example of writing coefficients elegantly with real numbers?

Sure, let's say we have the expression 2x + 3y. The coefficients in this expression are 2 and 3, which are attached to the variables x and y respectively. We can also write this expression as 2(x) + 3(y), making it more clear that the coefficients are being multiplied by the variables.

3. How do you determine the coefficients in an algebraic expression?

To determine the coefficients in an algebraic expression, you simply look at the number attached to the variable. If there is no number explicitly written, it is assumed to be 1. For example, in the expression 5x + 2y, the coefficients are 5 and 2.

4. Are there any rules or guidelines for writing coefficients elegantly with real numbers?

Yes, there are a few rules to keep in mind when writing coefficients with real numbers. First, coefficients should always be written before variables in an expression. Second, coefficients should be written as the smallest possible whole number. And lastly, coefficients should be separated from variables with a multiplication sign or parentheses.

5. How can writing coefficients elegantly with real numbers benefit us?

Writing coefficients elegantly with real numbers allows us to easily identify and understand the relationships between numbers in an expression. It also helps us to simplify and solve algebraic equations more efficiently. In addition, it can make mathematical expressions more visually appealing and easier to read.

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