# What is a coefficient of order unity?

• tony873004
In summary, the conversation discusses the concept of a coefficient of order unity and its role in a mathematical expression involving the Weierstrass's zeta function. The function is used to calculate a constant that is in the same order of magnitude as one. This constant is necessary to obtain an exact solution, as the given expression is only an approximation. The conversation also mentions the use of O(1) notation to represent values that remain finite as something else approaches a limit.

#### tony873004

Science Advisor
Gold Member
Does anyone know what a coefficient of order unity is? I'm reading a journal paper and it gives the formula
$$P_{KOZ} \simeq P_1 \left( {\frac{{m_0 + m_1 }}{{m_2 }}} \right)\left( {\frac{{a_2 }}{{a_1 }}} \right)^3 \left( {1 - e_2^2 } \right)^{3/2}$$
and then it says
This expression should be multiplied by a coefficient of order unity that can be obtained using Weierstrass's zeta function
.

It's on page 6 of this paper : http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v535n1/40691/406 [Broken] 91.web.pdf

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I'm sorry I can't help. Perhaps me bumping this up can get other people to see it additionally. However! Wikipedia is your friend!

This article seems to briefly explain the Weierstrass's zeta function.

$$\zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=1}^{\infty}\mathcal{G}_{2k+2}(\Lambda)z^{2k+1}$$

where $$\mathcal{G}_{2k+2}$$ is the Eisenstein series of weight $$2k+2$$.

So find the "coefficient of order unity" by using this function, then multiply it into whatever expression you're dealing with.

It means they are being sloppy and not calculating constants that go out front. These will be mathematical constants, like combinations of pi, e, etc, and will be not too small or too large, ie, probably in the range 1/10 to 10. So you might say what they have there is an approximation, although you should keep in mind that there is a constant that they could multiply it by that would make it exact for all values of whatever variable the function varies over, and this is something that isn't true in general of approximations.

tony873004 said:
Does anyone know what a coefficient of order unity is?
It may mean O(1) in the sense defined http://en.wikipedia.org/wiki/Big_oh" [Broken].

In short, the value remains finite as something else in the expression approaches some limit (e.g. infinity or zero depending on context).

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It means a small, unspecified constant in the same order of magnitude as one.

- Warren

chroot said:
It means a small, unspecified constant in the same order of magnitude as one.

- Warren
I guess that would imply that the answer given by the formula would be within an order of magnitude of correct without the unspecified constant.

Thanks everyone for your answers. I definitely learned a lot from this thread and the semi-related "What's the difference...?" thread.

## What is a coefficient of order unity?

A coefficient of order unity is a numerical value that is close to one, typically within a range of 0.5 to 1.5. It is used to describe the relative magnitude of a variable or quantity in a mathematical equation.

## How is a coefficient of order unity calculated?

A coefficient of order unity is calculated by dividing the value of the variable or quantity in question by its order of magnitude. For example, if the variable has a value of 100 and an order of magnitude of 10, the coefficient of order unity would be 10 (100/10 = 10).

## Why is a coefficient of order unity important in scientific research?

A coefficient of order unity is important because it helps to simplify and standardize mathematical equations, making them easier to understand and compare. It also allows scientists to focus on the most significant factors in their research and eliminate smaller, less significant variables.

## What is the significance of a coefficient of order unity being close to one?

A coefficient of order unity being close to one indicates that the variable or quantity has a high relative magnitude and is therefore an important factor in the equation. It also suggests that the variable has a strong influence on the outcome of the research or experiment.

## Can a coefficient of order unity be less than or greater than one?

Yes, a coefficient of order unity can be less than or greater than one, although it is most commonly found within a range of 0.5 to 1.5. A coefficient of order unity less than one indicates that the variable has a smaller relative magnitude, while a value greater than one indicates a larger relative magnitude.

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