Taylor Expansion for (a(1+z)^3 + b)^-1/2 around z=0 to First Order

In summary, the conversation discusses a function that needs to be Taylor expanded to the first order around z=0. The function has two constants a and b, which have a sum of one. The desired answer is 1 - (1+q)z, where q = a/2 - b. The person has tried to derive the answer themselves but is unsure if it is correct. The other person points out that the answer is likely correct based on the given information.
  • #1
indie452
124
0

Homework Statement



I have the function

(a(1+z)3 + b)-1/2

and i need to taylor expand it around z=0 to the first order,
a and b are constants, there sum is equal to one.

I have the answer:
1 - (1+q)z
where q = a/2 - b

This is in my physics book but it does not explain the steps inbetween. I have tried to derive it myself but i can't get same answer (or close to it), and I don't want to just accept it cause i might need to do it in the exam in the summer.
 
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  • #2
hi indie452! :smile:
indie452 said:
a and b are constants, there sum is equal to one.

I have the answer:
1 - (1+q)z
where q = a/2 - b

are you sure you have the wrong answer?

(1+q) = 1 + a/2 - b = 1 + a/2 - 1 + a = 3a/2 :wink:
 

FAQ: Taylor Expansion for (a(1+z)^3 + b)^-1/2 around z=0 to First Order

What is Taylor expansion around z=0?

Taylor expansion around z=0, also known as a Maclaurin series, is a mathematical method for approximating a function using a polynomial expression. It involves expanding a function into an infinite sum of terms, with each term representing a different degree of the function.

When is it useful to use Taylor expansion around z=0?

Taylor expansion around z=0 is useful for approximating the behavior of a function at a specific point, especially when the function is difficult to evaluate directly. It can also be used to find the derivatives of a function at a given point.

How do you perform Taylor expansion around z=0?

To perform Taylor expansion around z=0, you first need to find the derivatives of the function at the point z=0. Then, you can plug these values into the formula for a Taylor series to create an infinite sum of terms. The more terms you include, the more accurate your approximation will be.

What is the difference between Taylor expansion and Maclaurin series?

Taylor expansion and Maclaurin series are essentially the same thing. The only difference is that a Taylor series can be centered around any point, while a Maclaurin series is specifically centered around z=0.

What are the applications of Taylor expansion around z=0?

Taylor expansion around z=0 has many applications in mathematics, physics, and engineering. It can be used to approximate complicated functions, solve differential equations, and model real-world phenomena. It is also often used in optimization and numerical analysis.

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