Gary
GaryVasco
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Post by Gary on Jun 4, 2016 0:05:29 GMT
Vasco,
I don’t understand the second part of Ch 5, Ex 30 (ii), “Show that the first three terms in the binomial expansion of $S_C$ agree with those of S, ….” I looked at your document, but I don’t understand how one knows that “the binomial expansion of $S_C$” (line 2, p. 266) refers only to expansion of the square bracket. Is it just because that was the only component of the equation having the structure $(a + b)^n$? I believe I see the needed expansion procedure on p. 115.
Gary
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Post by Admin on Jun 5, 2016 14:33:57 GMT
Vasco, I don’t understand the second part of Ch 5, Ex 30 (ii), “Show that the first three terms in the binomial expansion of $S_C$ agree with those of S, ….” I looked at your document, but I don’t understand how one knows that “the binomial expansion of $S_C$” (line 2, p. 266) refers only to expansion of the square bracket. Is it just because that was the only component of the equation having the structure $(a + b)^n$? I believe I see the needed expansion procedure on p. 115. Gary Gary Notice that the Taylor series for $\mathcal{S}$ is a power series in $(z-a)$ and that the exercise talks about "in the vicinity of $a$" i.e. when $|(z-a)|$ is small. Although the Taylor series converges inside a circle whose radius is less than the distance to the nearest singularity, we only need a few terms of the series for a good approximation, if we choose $z$ such that $|(z-a|$ is small. Similarly the binomial expansion of the expression in square brackets is a power series in $(z-a)$ which converges when $|\mathcal{S}''/(2\mathcal{S}')(z-a)<1$. Using the result of Exercise 28 part (ii) we can see that $z$ must be inside the circle centred at $a$, with radius equal to the radius of the circle of curvature at $a$, for the binomial expansion to converge. So very generally speaking we are dealing with values of $z$ close to $a$. Also as you say we can see that the expression in square brackets is crying out for expansion, and so we give it a try! We then have a power series for $\mathcal{S}_C$ and $\mathcal{S}$, which we can compare term by term. As you say the required detail is on page 115. And we also need the hint given in part (ii) of the exercise. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Jun 5, 2016 18:17:31 GMT
Also as you say we can see that the expression in square brackets is crying out for expansion, and so we give it a try! We then have a power series for $S_C$ and $S$, which we can compare term by term. Vasco, Next time I will listen harder to the cries from within the square brackets. The explanation and comparison of the Taylor's and binomial series was very helpful. I wonder if one can make a general statement about when it is useful to express a function as one of these series. Is it always when we are most interested in values close to a point? Should we always expect the Taylor's series, by comparison to others, to give the best approximation when $\overset{.}{\kappa}$ is large, or is it just better than inversion in a circle under this circumstance? I'm trying to understand how one uses these series as tools for both complex analysis and for simulations. Gary |
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Post by Admin on Jun 6, 2016 12:38:01 GMT
Also as you say we can see that the expression in square brackets is crying out for expansion, and so we give it a try! We then have a power series for $S_C$ and $S$, which we can compare term by term. Vasco, Next time I will listen harder to the cries from within the square brackets. The explanation and comparison of the Taylor's and binomial series was very helpful. I wonder if one can make a general statement about when it is useful to express a function as one of these series. Is it always when we are most interested in values close to a point? Should we always expect the Taylor's series, by comparison to others, to give the best approximation when $\overset{.}{\kappa}$ is large, or is it just better than inversion in a circle under this circumstance? I'm trying to understand how one uses these series as tools for both complex analysis and for simulations. Gary Gary I think it would be difficult to make a general statement. What I would say by way of a summary is that any analytic function can be expressed as a Taylor series. This is shown in chapter 9 section II. It could be useful in many situations to be able to write a function as a power series, and under certain conditions, by taking a finite number of terms, an approximating polynomial is obtained. The binomial series, as it turns out, is a special case of Taylor's series when the function is of the form $[1+\alpha(z-a)]^p$, where $\alpha$ does not depend on $z$. The square brackets in exercise 30 are an example of this. So when we see something like $[c+b(z-a)]^p$ we can write it as $[c\{1+(b/c)(z-a)\}]^p$ and expand it according to the Binomial series rather than the more complicated (but equivalent) Taylor series route. Try it out with a simple function $[1+(z-a)]^p$. Expand it using the binomial theorem and the Taylor series about the point $a$ and you will get the same result. What the result of exercise 30 shows is that reflection in the tangent is a first order approximation (terms in the Taylor series upto $(z-a)$) and that inversion in C is a second order (better) approximation (terms in the series up to $(z-a)^2$). Part (iii) of the exercise shows that the error (difference between the full Taylor series and the inversion in C) is proportional to $\dot{\kappa}$ and so if the curvature of $K$ does not change much the error is small but if it changes a lot then the error will be larger in proportion to $\dot{\kappa}$. It would not be possible to do this type of error analysis without Taylor's series, which allows us to express a function as a power series. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Jun 6, 2016 16:25:32 GMT
Vasco, Next time I will listen harder to the cries from within the square brackets. The explanation and comparison of the Taylor's and binomial series was very helpful. I wonder if one can make a general statement about when it is useful to express a function as one of these series. Is it always when we are most interested in values close to a point? Should we always expect the Taylor's series, by comparison to others, to give the best approximation when $\overset{.}{\kappa}$ is large, or is it just better than inversion in a circle under this circumstance? I'm trying to understand how one uses these series as tools for both complex analysis and for simulations. Gary Gary I think it would be difficult to make a general statement. What I would say by way of a summary is that any analytic function can be expressed as a Taylor series. This is shown in chapter 9 section II. It could be useful in many situations to be able to write a function as a power series, and under certain conditions, by taking a finite number of terms, an approximating polynomial is obtained. The binomial series, as it turns out, is a special case of Taylor's series when the function is of the form $[1+\alpha(z-a)]^p$, where $\alpha$ does not depend on $z$. The square brackets in exercise 30 are an example of this. So when we see something like $[c+b(z-a)]^p$ we can write it as $[c\{1+(b/c)(z-a)\}]^p$ and expand it according to the Binomial series rather than the more complicated (but equivalent) Taylor series route. Try it out with a simple function $[1+(z-a)]^p$. Expand it using the binomial theorem and the Taylor series about the point $a$ and you will get the same result. What the result of exercise 30 shows is that reflection in the tangent is a first order approximation (terms in the Taylor series up to $(z-a)$) and that inversion in C is a second order (better) approximation (terms in the series up to $(z-a)^2$). Part (iii) of the exercise shows that the error (difference between the full Taylor series and the inversion in C) is proportional to $\dot{\kappa}$ and so if the curvature of $K$ does not change much the error is small but if it changes a lot then the error will be larger in proportion to $\dot{\kappa}$. It would not be possible to do this type of error analysis without Taylor's series, which allows us to express a function as a power series. Vasco Vasco, You have just answered questions that have been lingering in my mind for weeks, if not decades. And in this particular case and perhaps many other cases, the answer seems very practical as well. For accuracy and precision involving very wiggly curves, we want the Taylor series of the function of interest, assuming the function is convergent. If less accuracy and precision can be tolerated, a reasonable approximation can be obtained with simple reflection in the tangent or inversion in C in this case or with some other function in other cases, and we can determine the error by comparing higher order terms and, where possible, find comparable series by using the Binomial series (or perhaps some other easily calculated series?). This strategy of evaluation by comparing to the Taylor series should apply to many (all?) convergent analytic functions on curves and surfaces. Gary |
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