Riemann surface
matlab
还在面临数学物理方法的学习挑战吗？别担心！我们的mathematical-physics-methods团队专业为您解决各种与数学物理方法相关的问题。我们拥有深厚的专业背景和丰富的经验，能够帮助您完成高水平的作业和论文，让您的学习之路一帆风顺！
以下是一些我们可以帮助您解决的问题：
数学物理方法基础概念：涵盖各种常用的数学物理方法概念的定义、性质和分类，如微分方程、特殊函数、变分原理等。
数学物理模型：研究和应用于数学物理领域的模型和理论，如量子力学、统计物理、场论等。
数学物理方程求解：解析和数值求解数学物理方程的方法，如分离变量法、级数展开、有限元方法等。
数学物理优化问题：建模和求解与数学物理相关的优化问题，如最小化能量、最大化效率等。
数学物理计算方法：介绍数学物理方法在计算科学中的应用，如数值计算、数值优化等。
数学物理与量子力学：探讨数学物理方法在量子力学中的应用，如哈密顿算符、本征值问题等。
数学物理与统计物理：介绍数学物理方法在统计物理中的应用，如配分函数、统计力学等。
无论您面临的数学物理方法问题是什么，我们都会竭尽全力提供专业的帮助，确保您的学习之旅顺利无阻！
1.3 Investigate the properties of the polynomial equation
$$
f(x)=x^7+5 x^6+x^4-x^3+x^2-2=0,
$$
by proceeding as follows.
(a) By writing the fifth-degree polynomial appearing in the expression for $f^{\prime}(x)$ in the form $7 x^5+30 x^4+a(x-b)^2+c$, show that there is in fact only one positive root of $f(x)=0$.
(b) By evaluating $f(1), f(0)$ and $f(-1)$, and by inspecting the form of $f(x)$ for negative values of $x$, determine what you can about the positions of the real roots of $f(x)=0$.
(a) We start by finding the derivative of $f(x)$ and note that, because $f$ contains no linear term, $f^{\prime}$ can be written as the product of $x$ and a fifth-degree polynomial:
$$
\begin{aligned}
f(x) & =x^7+5 x^6+x^4-x^3+x^2-2=0, \
f^{\prime}(x) & =x\left(7 x^5+30 x^4+4 x^2-3 x+2\right) \
& =x\left[7 x^5+30 x^4+4\left(x-\frac{3}{8}\right)^2-4\left(\frac{3}{8}\right)^2+2\right] \
& =x\left[7 x^5+30 x^4+4\left(x-\frac{3}{8}\right)^2+\frac{23}{16}\right] .
\end{aligned}
$$
Since, for positive $x$, every term in this last expression is necessarily positive, it follows that $f^{\prime}(x)$ can have no zeros in the range $0<x<\infty$. Consequently, $f(x)$ can have no turning points in that range and $f(x)=0$ can have at most one root in the same range. However, $f(+\infty)=+\infty$ and $f(0)=-2<0$ and so $f(x)=0$ has at least one root in $0<x<\infty$. Consequently it has exactly one root in the range.
(b) $f(1)=5, f(0)=-2$ and $f(-1)=5$, and so there is at least one root in each of the ranges $0<x<1$ and $-1<x<0$.
There is no simple systematic way to examine the form of a general polynomial function for the purpose of determining where its zeros lie, but it is sometimes
helpful to group terms in the polynomial and determine how the sign of each group depends upon the range in which $x$ lies. Here grouping successive pairs of terms yields some information as follows:
$x^7+5 x^6$ is positive for $x>-5$,
$x^4-x^3$ is positive for $x>1$ and $x<0$, $x^2-2$ is positive for $x>\sqrt{2}$ and $x<-\sqrt{2}$. Thus, all three terms are positive in the range(s) common to these, namely $-51$. It follows that $f(x)$ is positive definite in these ranges and there can be no roots of $f(x)=0$ within them. However, since $f(x)$ is negative for large negative $x$, there must be at least one root $\alpha$ with $\alpha<-5$.
1.5 Construct the quadratic equations that have the following pairs of roots:
(a) $-6,-3 ;$; (b) 0,4 ; (c) 2,$2 ;$; (d) $3+2 i, 3-2 i$, where $i^2=-1$.
Starting in each case from the ‘product of factors’ form of the quadratic equation, $\left(x-\alpha_1\right)\left(x-\alpha_2\right)=0$, we obtain:
(a) $(x+6)(x+3)=x^2+9 x+18=0$;
(b) $(x-0)(x-4)=x^2-4 x=0$;
(c) $(x-2)(x-2)=x^2-4 x+4=0$;
(d) $(x-3-2 i)(x-3+2 i)=x^2+x(-3-2 i-3+2 i)$
$$
\begin{gathered}
+\left(9-6 i+6 i-4 i^2\right) \
=x^2-6 x+13=0
\end{gathered}
$$
E-mail: help-assignment@gmail.com 微信:shuxuejun
help-assignment™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注于数学，统计，金融，经济，计算机科学，物理的作业代写服务