Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Textbook Question

Chapter 7.12, Problem 12P

In Problems 3 to 12, find the exponential Fourier transform of the given f ( x ) and write f ( x ) as a Fourier integral [that is, find g ( α ) in equation ( 12.2 ) and substitute your result into the first integral in equation (12.2)].

f ( x ) = { sin x , | x | < π / 2 0 , | x | > π / 2

Hint: In Problems 11 and 12, use complex exponentials.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.12, Problem 11P

Chapter 7.12, Problem 13P

Students have asked these similar questions

Module Code: MATH380202 3. (a) Let {} be a white noise process with variance σ2. Define an ARMA(p,q) process {X} in terms of {+} and state (without proof) conditions for {X} to be (i) weakly stationary and (ii) invertible. Define what is meant by an ARIMA (p, d, q) process. Let {Y} be such an ARIMA(p, d, q) process and show how it can also be represented as an ARMA process, giving the AR and MA orders of this representation. (b) The following tables show the first nine sample autocorrelations and partial auto- correlations of X and Y₁ = VX+ for a series of n = 1095 observations. (Notice that the notation in this part has no relationship with the notation in part (a) of this question.) Identify a model for this time series and obtain preliminary estimates for the pa- rameters of your model. X₁ = 15.51, s² = 317.43. k 1 2 3 4 5 6 7 Pk 0.981 0.974 0.968 akk 0.981 0.327 8 9 0.927 0.963 0.957 0.951 0.943 0.935 0.121 0.104 0.000 0.014 -0.067 -0.068 -0.012 Y₁ = VX : y = 0.03, s² = 11.48. k 1…

Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.

Module Code: MATH380202 1. (a) Define the terms "strongly stationary" and "weakly stationary". Let {X} be a stochastic process defined for all t € Z. Assuming that {X+} is weakly stationary, define the autocorrelation function (acf) Pk, for lag k. What conditions must a process {X+) satisfy for it to be white noise? (b) Let N(0, 1) for t€ Z, with the {+} being mutually independent. Which of the following processes {X+} are weakly stationary for t> 0? Briefly justify your answers. i. Xt for all > 0. ii. Xo~N(0,) and X₁ = 2X+-1+ &t for t > 0. (c) Provide an expression for estimating the autocovariance function for a sample X1,..., X believed to be from a weakly stationary process. How is the autocor- relation function Pk then estimated, and a correlogram (or acf plot) constructed? (d) Consider the weakly stationary stochastic process ✗+ = + + +-1+ +-2 where {E} is a white noise process with variance 1. Compute the population autocorre- lation function Pk for all k = 0, 1, ....

Chapter 7 Solutions

Mathematical Methods in the Physical Sciences

Additional Math Textbook Solutions

Find more solutions based on key concepts

50. Swimming Recently, a group of adults who swim regularly for exercise were evaluated for depression. It turn...

Stats

3. For the same sample statistics, which level of confidence would produce the widest confidence interval? Expl...

Elementary Statistics: Picturing the World (7th Edition)

For each of the following, determine the constant c so that f(x) satisfies the conditions of being a pmf for a ...

Probability And Statistical Inference (10th Edition)

2. Source of Data In conducting a statistical study, why is it important to consider the source of the data?

Elementary Statistics

ASSESSMENT Find the first five terms in sequences with the following nth terms. a. n2+2 b. 5n+1 c. 10n1 d. 3n2 ...

A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)

Using the Central Limit Theorem. In Exercises 5–8, assume that females have pulse rates that are normally distr...

Elementary Statistics (13th Edition)

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.