What is Taylor expansion: Definition and 174 Discussions
In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).
How do you Taylor expand e^{i \vec{k} \cdot \vec{r}}
the general formula is \phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{a})
but \vec{k} \cdot \vec{r} isn't of the form \vec{r}+\vec{a} is it?
Homework Statement
I'm having a hard time following a taylor expansion that contains vectors...
http://img9.imageshack.us/img9/9656/blahz.png
http://g.imageshack.us/img9/blahz.png/1/
Homework Equations
The Attempt at a Solution
Here's how I would expand it:
-GMR/R^3 -...
Homework Statement
f(E) = \left(\frac{E_c}{E} \right)^{1/2} + \frac{E}{kT}
Expand this as a Taylor function with the form...
f \approx a_0 + a_1(E-E_0) + a_2(E-E_0)
Hint being a_1 will be 0, because E_0 is a Gamow peak in this case, so slope will be 0.
What I need to do is...
Homework Statement
Develop the Taylor expansion of ln(1+z).
Homework Equations
Taylor Expansion: f(z) = sum (n=0 to infinity) (z-z0)n{f(n)(z0)}/{n!}
Cauchy Integral Formula: f(z) = (1/(2*pi*i)) <<Closed Integral>> {dz' f(z')} / {z'-z}
The Attempt at a Solution
I have NO idea...
Homework Statement
With n>1, show that (a) \frac{1}{n}-ln\frac{n}{n-1}<0
and (b) \frac{1}{n}-ln\frac{n+1}{n}>0
Use these inequalities to show that the Euler-Mascheron constant (eq. 5.28 - page330) is finite.
Homework Equations
This is in the chapter on infinite series, in the section...
Homework Statement
What is the quadratic approximation to the potential function?
Homework Equations
U(x) = U0((a/x)+(x/a))
U0= 20
a=4
The Attempt at a Solution
This is just the last part of a question on my engineering homework, I never learned Taylor expansions before even...
Hi There. Was working on these and I think I managed to get most of them but still have a few niggling parts. I've managed to do questions 2,3,3Part2 and I've shown my working out so I'd be greatful if you could verify whether they are correct.
Please could you also guide me on Q1 & 4. Q1...
Homework Statement
I need to find the bloch vector for the density matrix \frac{1}{N}\exp{-\frac{H}{-k_bT}} where the Hamiltonian is given by H=\hbar\omega\sigma_z.
The Attempt at a Solution
I can break the Taylor series of exp into odd and even terms because sigma z squared is the...
Homework Statement
For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 %
Homework Equations
The taylor series from 0 to 1 , the linearization, is the most appropriate equation
The Attempt at a Solution...
Homework Statement
Linearize the system operator illustrated below by applying a Taylor series expansion.
f(t) ----> e^f(t) -----> g(t)
Homework Equations
I only find the general form of a taylor series relevant.
g(x)= sum (0,infinity) of [f^n*(a)*(x-a)^n]/n!
The system is...
This is just part of a larger problem, but I have a basic equation r'=k-g*r, where k and a start out as constants, but then I need to treat everything as if it can vary slightly from the average. For this, I set r=r_ave+dr, g=g_ave+dg, and k=k_ave+dk. Now I need to work these into the first...
Hi I wonder if there is a simpler way to obtain the first three non-zero terms of Taylor Expansion for the function \frac{Ln(1+x)}{1-x} about x=0?
I differentiated it directly, but it was such a nightmare to do:mad: . So I am wondering if there is a simpler way to do it?
Homework Statement
I have the following question to answer:
Show that
(X^2/h^2)*((1/2*y1) - y2 + (1/2*y3)) + (X/h)*((-1/2 y1)+(1/2 y3))+y2 (sorry about the format)
is equal to (taylor expansion):
y = y2+(x(dy/dx)¦0 + (x^2/2*((d^2)y)/(dx^2))¦0
Homework Equations
also given in...
Hey Everyone.
I'm ALMOST finished this problem...
To spare you the long story, I need to take the difference between an gravitational acceleration, and the same gravitational acceleration at a slightly larger height.
The two functions are a(r) and a(r+d), where d is very small
Now... VERY...
I'm having trouble determining the order of the pole of
[exp(iz) - 1]/((z^2) + 4) at z=2i
I know I can't just expand the exponential as 1 + iz + [(iz)^2]/2 ...
because this formula only works near the origin. Can I still use Taylor's theorem to find the expansion at z=2i (i.e does...
the problem reads develop expansion of ln(1+z)
of course I just tried throwing it into the formula for taylor expansions, however I do not know what F(a) is, the problem doesn't specify, so how can I use a taylor series?
Can anyone please give me a hint on any of the following Taylor expansions? That would be so helpful!
for all three: Find the first non-zero term in the Taylor series about x = 0
problem 1
\frac{1} {sin^2x} - \frac{1} {x^2}
everytime I differentiate the result is zero...so that...
I am supposed to find an approximation of this integral evaluated between the limits 0 and 1 using a taylor expansion for cos x:
\int \frac{1 - cos x}{x}dx
and given
cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}...
i should get a simple series similar to this for...
V = 2\pi \sigma(\sqrt{R^2+a^2}-R)
Show that for large R,
V \approx \frac{\pi a^2 \sigma}{R}
I figured if I could develop the MacLaurin serie with respect to an expression in R such that when R is very large, this expression is near zero, then the first 1 or 2 terms should be a fairly...
basic taylor expansion...
Hi, could some one explain how i could use the taylor series to expand out:
f(x)= 1/sqrt(1-x^2)
Any help would be appreciated, thanks.