Continuous and Derivative is Zero Then Constant

Fundamental Theories and Mechanisms of Failure

K. Ravi-Chandar , in Comprehensive Structural Integrity, 2003

2.05.2.5 Propagation of Discontinuities: Wavefronts and Rays

The displacement vector $\mathbf{u}\left(\mathbf{x},t\right)$ need not possess continuous derivatives; the governing equation (4), allows discontinuities in the derivatives of $\mathbf{u}\left(\mathbf{x},t\right)$ to exist along certain planes (called wavefronts) and to propagate along certain directions (called rays). Discontinuities in the spatial gradients of $\mathbf{u}\left(\mathbf{x},t\right)$ imply a discontinuity in the strains and stresses and discontinuities in the temporal gradients of $\mathbf{u}\left(\mathbf{x},t\right)$ indicate jumps in the particle velocity and/or acceleration. While such discontinuities cannot be sustained physically, rapid changes in $\mathbf{u}\left(\mathbf{x},t\right)$ that occur over a very short distances or time intervals are approximated as discontinuous jumps in the gradients. This representation is useful in characterizing the variations in the strains, stresses, and velocities generated by suddenly applied loads. Love (1927). described the kinematic and dynamic conditions that must hold on a surface of discontinuity. With the normal to the discontinuity denoted by n , the kinematic and dynamic jump conditions are

(23) $\left[{\dot{u}}_i\right]=-c\left[u_{i,j}\right]n_j$

(24) $-\rho c\left[{\dot{u}}_i\right]=\left[{\sigma }_{\mathit{ij}}\right]n_j$

where ρ is the density, c is the appropriate wave speed, and the square bracket around a quantity indicates the jump in that quantity across the discontinuity. Equations (23) and (24) can be interpreted using the equations of motion (4). Introducing (4) into (24) yields

(25) $\rho c\left[{\dot{u}}_i\right]=-\lambda {\delta }_{\mathit{ij}}\left[u_{k,k}\right]n_j-\mu \left[u_{i,j}\right]n_j-\mu \left[u_{j,i}\right]n_j$

which may be rearranged as follows:

(26) $\left(\rho c^2-\mu \right)\left[{\dot{u}}_i\right]=-c\left[\lambda +\mu \right]\left[u_{k,k}\right]n_j$

If we impose a velocity jump $\left[{\dot{u}}_i\right]$ , with zero dilatation $\left[u_{kk}\right]=0$ , the jump propagates at a speed $c=C_{\mathrm{s}}$ . In a similar manner, if we consider a velocity jump $\left[{\dot{u}}_i\right]$ with $\nabla \times \mathbf{u}=0$ , Equation (25) reduces to

(27) $\left(\rho c^2-\left(\lambda +2\mu \right)\right)\left[{\dot{u}}_i\right]n_i=0$

which indicates that jumps in dilatation travel with the speed $c=C_{\mathrm{d}}$ . Therefore, we might expect that if an arbitrary velocity jump is provided (through an external loading agent or from an internal source), both dilatational and shear waves propagate in the body carrying the appropriate jump discontinuities along both wavefronts.

The construction of wavefronts and rays is useful in understanding and interpreting the development of stress fields in elastodynamic problems. So, we shall briefly outline the construction of the equations for the wavefronts and rays. The surface of discontinuity may be written as $S\left(x,t\right)=\tau \left(\mathbf{x}\right)-t=0$ or equivalently by $t=\tau \left(\mathbf{x}\right)$ . At any point on the wavefront, the ray is normal to the wavefront; thus, the governing equation for the rays is obtained:

(28) $\frac{\mathrm{d}\mathbf{x}}{\text{d}t}=c\mathbf{n}=c\frac{\nabla \tau \left(\mathbf{x}\right)}{\left|\nabla \tau \left(\mathbf{x}\right)\right|}$

but

$\frac{\text{d}S}{\text{d}t}=\frac{\partial \tau \left(x\right)\partial x_i}{\partial x_i\partial t}-1=0\text{or}c\nabla \tau \left(x\right)\cdot n=1$

Using the second expression in Equation (28) and the above, we get the equation for the ray as

(29) $c\nabla \tau \left(\mathbf{x}\right)=1$

This is called the eikonal equation, the terminology arising from geometrical optics. Using (29) in (28) results in the following equation for the rays:

(30) $\frac{\mathrm{d}\mathbf{x}}{\text{d}t}=c^2\nabla \tau \left(\mathbf{x}\right)$

In the linearly elastic solid, the wave speeds are constant and therefore the rays are straight lines and the corresponding wavefronts are parallel surfaces. Therefore, the knowledge of the wavefront at some time t can be used to construct the wavefront at a later time $t^{\prime }$ simply by extending the rays along the normal to the wavefront by amount $c(t^{\prime }-t)$ . In the optics literature, this is called Huygens' principle and Huygens' construction of wavefronts. This construction is quite useful in obtaining a quick, qualitative picture of the wave propagation event as we shall see in later examples.

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On the Convergence of Discrete Approximations to the Navier-Stokes Equations*

Alexandre Joel Chorin , in Computational Fluid Mechanics, 1989

THEOREM 3.

Let Eqs. (1) and (2) have a periodic solution with continuous derivatives up to order five for 0 ≦ t ≦ T. Let k = O(h ²); if ||u ⁰ – w ⁰||, k and h are sufficiently small, Eqs. (14) have a unique solution which converges to the solution of (1) and (2) in both the L ₂ and maximum norms. The error in the L ₂ norm is of order h ²; the error in the maximum norm is bounded by O(h) in the two-dimensional case and by O(√h) in the three-dimensional case.

Theorem 3 and its proof can be summarized as follows: Let u _z ⁿ , w _z ⁿ be vector functions defined for z in Ω _h and for n such that 0 ≦ nk ≦ T ₁; introduce the "space-time" maximum and L ₂ norms

$\begin{array}{l}{\Vert \text{u}\Vert }_{\max ,T_1}=\underset{0\leqq nk<T_1}{\max }{\Vert {\text{u}}^n\Vert }_{\max }\\ {\Vert \text{u}\Vert }_{T_1}=\underset{0\leqq nk<T_1}{\max }\Vert {\text{u}}^n\Vert .\end{array}$

The equations

$\begin{array}{l}(I-k{Q}_1({\omega }^n)){\text{u}}^{n+1/3}={\text{u}}^n,\\ (I-k{Q}_2({\omega }^n)){\text{u}}^{n+2/3}={\text{u}}^{n+1/3},\\ {\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}u}}^{n+1}=P({\text{u}}^{n+2/3}+k{\text{E}}^{n+1}),\\ u^{0}ugiven,\end{array}$

define a mapping ω → u. This mapping maps the maximum norm sphere

${\Vert \text{ω}\Vert }_{\max ,T_1}\leqq 2{\Vert \text{w}\Vert }_{\max ,T_1}$

into the L ₂ norm sphere

${\Vert \text{u}-\text{w}\Vert }_{T_1}\leqq {\Vert \text{w}\Vert }_{\max ,T_1}{h}^{1/2}.$

For ||u ⁰ – w ⁰||, k and h sufficiently small, this mapping has a unique fixed point which is the solution of (14) and lies close to v, the solution of (1) and (2).

In our analysis we have neglected the effect of round-off error and of the errors arising from the possibly incomplete iterative evaluation of u ⁿ⁺¹ = P(u ^n+2/3 + k E ⁿ⁺¹). It is obvious, however, that the analysis remains valid if the round-off errors are of order k ² and provided u ⁿ⁺¹ is approximated by a vector (u ⁿ⁺¹)^* such that D(u ⁿ⁺¹)^* = O(k ²). Furthermore, in the dimensionless variables used in this paper the effect of the Reynolds number R on the error is not in evidence. Clearly C depends on R and increases as R increases; i.e. as R increases k and h have to be reduced for accuracy to be preserved. Finally, it is clear that the results of this section apply to certain other quasi-linear equations besides the Navier-Stokes equations, provided the boundary conditions are homogeneous. In this sense, our results generalize the work of M. Lees (see e.g. [8]), who considered equations with nonlinear terms of a simpler nature.

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ON THE SOLUTIONS OF SECOND AND THIRD ORDER DIFFERENTIAL EQUATIONS

Shoshana Abramovich , in Dynamical Systems, 1977

Under these conditions

when p(x), f(x) and r(x) are positive, having continuous derivative, p(x)r(x) is increasing and f(x)r(x) is decreasing, we get that the zeros of a solution y ₁ (x) of equation (1) and a solution y _a (x) of equation (2), defined by y ₁ (0) = y _a (o) = 0, y′ _a (0) = a, seperate each other, moreover, the zeros of y _a (x) for different values of a seperate each other [1, T.h. 1].

By making the same assumptions as before we get that between two zeros of a solution of (2) there is only one extremum [1, Th. 2].

Under these restrictions that: p(x), f(x), r(x) are positive, p(x)r(x) is nondecreasing, and f(x)/p(x) is a nonincreasing function we get that the maxima of solutions of (2) are decreasing [1, Th. 3].

Parts of the above theorems about monotone functions can be extended to a larger class of functions, for instance, to a class of functions for which the condition p(a-x) ≥ p(a+x), x > 0, (such p(x) is called left balanced with respect to a).

As for the third order linear equation (3), under the conditions when: r(x) > 0, r′(x) ≤ 0, and q(x) ≥ 0 are continuous functions for x ≥ 0, and under the condition when r(x)p(x) is a continuous increasing function we get, that a nontrivial solution of (3) defined by y(0) = α, y′(0) = 0, y′(0) = β, α, β ≥ 0, is positive and its minima are increasing for x ≥ 0. [2, Th. 1].

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Damping

Alvar M. Kabe , Brian H. Sako , in Structural Dynamics Fundamentals and Advanced Applications, 2020

Appendix 4.1 Taylor series expansion

A function $f\left(\zeta \right)$ can be expressed by a Taylor power series expansion provided it has an $n$ th continuous derivative at a (Crowell and Slesnick, 1968), i.e.,

$f\left(\zeta \right)=f\left(a\right)+f^{\prime }\left(a\right)(\zeta -a)+\cdots +\frac{1}{n!}{f}^{\left(n\right)}\left(a\right){(\zeta -a)}^n+R_n$

We seek such an expansion for the following function:

$f\left(\zeta \right)={\left(1+\zeta \right)}^{-1/2}+{\left(1-\zeta \right)}^{-1/2}$

The first five terms in the series are

$\begin{array}{l}f\left(0\right)={\left(1+0\right)}^{-1/2}+{\left(1-0\right)}^{-1/2}=2\\ f^{\prime }\left(0\right)=-\frac{1}{2}\left\{{\left(1+0\right)}^{-3/2}-{\left(1-0\right)}^{-3/2}\right\}=0\\ f^{″}\left(0\right)=\frac{3}{4}\left\{{\left(1+0\right)}^{-5/2}+{\left(1-0\right)}^{-5/2}\right\}=\frac{3}{4}\\ f^{\left(3\right)}\left(0\right)=-\frac{15}{8}\left\{{\left(1+0\right)}^{-7/2}-{\left(1-0\right)}^{-7/2}\right\}=0\\ f^{\left(4\right)}\left(0\right)=\frac{105}{16}\left\{{\left(1+0\right)}^{-9/2}+{\left(1-0\right)}^{-9/2}\right\}=\frac{105}{8}\end{array}$

Substituting into the Taylor series expansion we obtain

$\begin{array}{c}f\left(\zeta \right)=2+0+\left(\frac{1}{2!}\right)\left(\frac{3}{4}\right){\left(\zeta -0\right)}^2+0+\left(\frac{1}{4!}\right)\left(\frac{105}{8}\right){\left(\zeta -0\right)}^4+R_n\\ =2\left(1+\frac{3}{16}{\zeta }^2+\frac{35}{128}{\zeta }^4+R_n\right)\end{array}$

For lightly damped structures, $\zeta$ will be small and we can neglect its higher-order terms, which then yields

$f\left(\zeta \right)={\left(1+\zeta \right)}^{-1/2}+{\left(1-\zeta \right)}^{-1/2}\approx 2$

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LINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS ON A BANACH SPACE

Richard Datko , in Dynamical Systems, 1977

1 Theorem

The domain of a consists of those ϕ∈C[[-h,0], X] which posses continuous derivatives, for which ϕ(0) ε D(A) and which satisfy the relation

(4) $\varphi \left(0\right)={\displaystyle \sum _{\text{j}=1}^{\text{m}}{\text{B}}_{\text{j}}\varphi \left(-{\text{h}}_{\text{j}}\right)}+\text{A}\varphi \left(0\right)+{\displaystyle \sum _{\text{j}=1}^{\text{m}}{\text{A}}_{\text{j}}\varphi \left(-{\text{h}}_{\text{j}}\right)}.$

If ϕ is in the domain of a then the solution of (2) satisfies the differential equation (1).

The next theorem describes a representation for the solutions of (2) which is the analogue of the one found in the finite dimensional case.

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Signal and Image Representation in Combined Spaces

Ami Harten , in Wavelet Analysis and Its Applications, 1998

§6 ENO reconstruction and subcell resolution

In [19] we presented a data-dependent piecewise-polynomial reconstruction technique which avoids the Gibbs phenomenon by an adaptive selection of stencil $S_{〉}^{\left|\right|}$ in (3.9); we refer to this technique as Essentially Non-Oscillatory (ENO) reconstruction. The basic idea of ENO reconstruction is to assign to a cell c _i ^k which is in the smooth part of the sampled function, a stencil

with i ₀= i ₀(i) which is likewise in the smooth part of the function (provided that this is possible, i.e. that discontinuities are well separated and are far enough from the boundaries). This is done by choosing $S_{〉}^{\left|\right|}$ to be the stencil for which the reconstruction polynomial p _i ^k (x;

v ^k ) in (3.9) is the "smoothest" among all candidate stencils, i.e. those of r consecutive cells of C^k (starting with $c_{{}_{i_0}}^k$ ) which contain the cell c _i ^k , e.g. by taking i ₀(i) to be the index for which

(6.1) $\underset{i_0}{min}\left|\frac{d^{r-1}}{d{x}^{r-1}}{p}_i^k\left(x;v^k\right)\right|$

is attained among all candidate stencils. This enables us to get a good approximation everywhere except in the cells which contain a discontinuity.

Next we show that cell-average discretization enables us to get a good approximation even in cells which contain discontinuity by using "subcell resolution" (see [15]). Let ℐ _k (x; F ^k ) denote the piecewise-polynomial interpolation of the primitive function in (3.9). Since it has formal order of accuracy r + 1, we get in regions of smoothness of f that

(6.2a) ${\mathrm{ℐ}}_{\left|\right|}\left(\S ;{\mathrm{ℱ}}^{\left|\right|}\right)=\mathrm{ℱ}\left(\S \right)+\mathcal{O}\left(\right(〈{}_{\left|\right|}\left){}^{\nabla +\infty }\right|\left|{\mathrm{ℱ}}^{\left(\nabla +\infty \right)}\right|\left|\right)$

then

(6.2b) $\begin{array}{l}\left({\mathrm{ℛ}}_k{v}^k\right)\left(x\right)=\frac{d}{dx}{I}_k\left(x;F^k\right)=\frac{d}{dx}F\left(x\right)+O\left({\left(h_k\right)}^r\left|\right|F^{\left(r+1\right)}\left|\right|\right)\\ \begin{array}{cccc}\hfill \begin{array}{cccc}\hfill \begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \end{array}\hfill & \hfill \hfill & \hfill \hfill & \hfill =f\left(x\right)+O\left(\right(h_k\hfill \end{array}\left){}^r\right|\left|f^{\left(r\right)}\right|\left|\right)\text{.}\end{array}$

Assume now that f(x) has (p – 1) continuous derivatives and that f ^(p)(x) is discontinuous but bounded. It is clear from relations (6.2) that the maximal accuracy that can be achieved from either point values or cell averages is $O\left(h^p\left|\right|f^{\left(p\right)}\left|\right|\right):$ Using cell-averages we gain one order of smoothness in the primitive function (3.5a) but we lose it in the differentiation (3.5b). Consequently there is no advantage in using cell averages rather than point values of f(x) for smooth data.

There is a significant advantage however in using cell averages rather than point values of f when f(x) is discontinuous in a finite number of points. To see that, let us assume that f(x) is discontinuous at $x_d\in \left(x_{j-1}^k,x_j^k\right)$ and that in [a, x_d ) ∪ (x_d ,b], 0 ≤ a < x_d < b ≤ l, f has at least r continuous derivatives. Let ℐ^ℒ and ℐ^ℛ denote interpolation of either f(x) or F(x) at grid points in [a, x_d ) and (x_d , b], respectively. We note that F(x) is continuous in [a, b], but has a discontinuous derivative at x_d . Consequently, if F(x) is properly resolved on the k-th grid ${\mathrm{ℐ}}^{\mathrm{ℒ}}\left(\S ;{\mathrm{ℱ}}^{\left|\right|}\right)$ and ${\mathrm{ℐ}}^{\mathrm{ℛ}}\left(\S ;{\mathrm{ℱ}}^{\left|\right|}\right)$ will intersect at some point ${\tilde{x}}_d\in c_j^k$ From (6.2) we get that this intersection point is a good approximation to the location of the discontinuity within the cell c _j ^k , i.e.

(6.3) ${\tilde{x}}_d-x_d=O\left({\left(h_k\right)}^r\left|\right|f^{\left(r\right)}\left|\right|\right))$

On the other hand, having knowledge of point-values {f(x _i ^k )} in [a, b], there is nothing much we can say about the location of the discontinuity within the cell c _j ^k

We describe now how to use the subcell-resolution technique of [15] in the prediction (3.8) in order to get an accurate prediction in a cell $c_j^{k-1}$

which contains a discontinuity. Let us denote by ${\tilde{F}}_{2j-1}^k$ our approximation to $F\left(x_{2j-1}^k\right)$ and let ${\mathrm{ℐ}}^{\mathrm{ℒ}}\left(\S ;{\mathrm{ℱ}}^{\left|\right|-\infty }\right)$ and ${\mathrm{ℐ}}^{\mathrm{ℛ}}\left(\S ;{\mathrm{ℱ}}^{\left|\right|-\infty }\right)$ denote the ENO interpolation in the neighboring cells $c_{j-1}^{k-1}$ and $c_{j+1}^{k-1}$ on the left and right, respectively, and define

(6.4a) $D\left(x\right)={\mathrm{ℐ}}^{\mathrm{ℛ}}\left(\S ;{\mathrm{ℱ}}^{\left|\right|-\infty }\right)-{\mathrm{ℐ}}^{\mathrm{ℒ}}\left(\S ;{\mathrm{ℱ}}^{\left|\right|-\infty }\right)\text{.}$

Since $D\left({\tilde{x}}_d\right)=0$ we assume that

(6.4b) $D\left(x_{j-1}^{k-1}\right).D\left(x_j^{k-1}\right)<0$

${\tilde{F}}_{2j-1}^k$ is now computed as follows

(6.5) ${\tilde{F}}_{2j-1}^k=\left\{\begin{array}{c}\hfill {\mathrm{ℐ}}^{\mathrm{ℒ}}\left({\S }_{\in |-\infty }^{\left|\right|};{\mathrm{ℱ}}^{\left|\right|-\infty }\right)\mathrm{if}D\left(x_{2j-1}^k\right)\cdot D\left(x_j^{k-1}\right)\le 0\hfill \\ \hfill {\mathrm{ℐ}}^{\mathrm{ℛ}}\left({\S }_{\in |-\infty }^{\left|\right|};{\mathrm{ℱ}}^{\left|\right|-\infty }\right)\mathit{otherwise}\text{.}\hfill \end{array}\right.$

It is easy to see that if f(x) is a piecewise-polynomial function

(6.6a) $f\left(x\right)=\left\{\begin{array}{c}\hfill P_L\left(x\right)a\le x<x_d\hfill \\ \hfill P_R\left(x\right)x_d<x\le b\hfill \end{array}\right.$

with

(6.6b) $deg\left(P_L\right)\le r-1,deg\left(p_R\right)\le r-1\text{,}$

then

(6.6c) ${\tilde{F}}_{2j-1}^k=F\left(x_{2j-1}^k\right)\text{,}$

i.e. the procedure (6.5) is exact. More generally, if f(x) has p continuous derivatives to the left and the right of the discontinuity, we get that

(6.7) ${\tilde{F}}_{2j-1}^k=F\left(x_{2j-1}^k\right)+O\left({\left(h_k\right)}^{\overline{p}+1}\left|\right|f^{\left(\overline{p}\right)}\left|\right|\right),\overline{p}=min\left(p,r\right)\text{.}$

Remark 2

If we know that f(x) has q – 1 continuous derivatives and a discontinuity of the q-th derivative in $x_d,x_{j-1}^{k-1}<x_d<x_j^{k-1}$ we can extend the subcell resolution technique of (6.4)-(6.5) to this case as follows: $\frac{d^q}{d{x}^q}F\left(x\right)$ has a discontinuous first derivative at x_d If it is sufficiently resolved on the grid, we expect $\frac{d^q}{d{x}^q}{I}_L\left(x;F^{k-1}\right)$ and $\frac{d^q}{d{x}^q}{I}_k\left(x;F^{k-1}\right)$ to intersect at id in $c_j^{k-1}$ ,

(6.8a) ${\tilde{x}}_d-x_d=O\left(h^{r-q}\right)\text{.}$

It follows, therefore, that if we replace D(x) in (6.4) by

(6.8b) $D\left(x\right)=\frac{d^q}{d{x}^q}{I}^R\left(x;F^{k-1}\right)-\frac{d^q}{d{x}^q}{I}^L\left(x;F^{k-1}\right)\text{,}$

we get a subcell-resolution technique which is exact for the corresponding piecewise-polynomial problem (6.6); this implies (6.7).

Remark 3

Extrapolating the analysis of the information contents in cell averages vs. point values, we get that weighted averages with respect to the hat-function contain information that will enable us to obtain subcell resolution of δ-distributions; this may be useful for compression of digital images and propagation of singularities (see [2]).

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Topics of Functional Analysis

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

18.1 Linear and normed spaces of functions

Below we will introduce examples of some functional spaces with the corresponding norm within. The lineality and main properties of a norm (metric) can be easily verified that is why we leave this for the reader as an exercise.

18.1.1 Space m_n of all bounded complex numbers

Let us consider a set m of sequences $x:={\left\{x_i\right\}}_{{}_{i=1}}^{{}^{\infty }}$ such that

where $\left|\right|x_i\left|\right|:={\sqrt{{\displaystyle {\sum }_{s=1}^n{x}_{is}\overline{x}}}}_{is}$ and introduce the norm in m as

18.1.2 Space lⁿ _p of all summable complex sequences

By definition

(18.4) $l_p^n:=\left\{\begin{array}{cc}x={\left\{x_i\right\}}_{i=1}^{\infty }|x_i\in C^n,& {\Vert x\Vert }_{l_p^n}:={({\displaystyle \sum }_{i=1}^{\infty }{\Vert x_i\Vert }^p)}^{1/p}\end{array}\right\}<\infty$

18.1.3 Space C[a, b] of continuous functions

It is defined as follows

(18.5) $\begin{array}{ll}C[a,b]:=\hfill & \begin{array}{ccc}\{f(t)|f& \text{is continuous for all}& t\in [a,b],\end{array}\hfill \\ \hfill & {\Vert f\Vert }_{C[a,b]}:=\underset{t\in [a.b]}{\max }|f(t)|<\infty \}\hfill \end{array}$

18.1.4 Space C^k [a, b] of continuously differentiable functions

It contains all functions which are k-times differentiable and the kth derivative is continuous, that is,

(18.6) $\begin{array}{rrr}\hfill & \hfill C^k[a,b]:=& \hfill \{f(t)|f^{(k)}\\ \hfill \text{for all}& \hfill t\in [a,b],& \hfill {\Vert f\Vert }_{C^k[a,b]}\end{array}\begin{array}{c}\text{ exists and is continuous}\\ :={\displaystyle \sum }_{i=0}^k\underset{t\in [a.b]}{\max }|f^{(i)}(t)|<\infty \}\end{array}$

18.1.5 Lebesgue spaces $L_p\left[a,b\right]\left(1\le p<\infty \right)$

For each $1\le p<\infty$ it is defined by the following way:

(18.7) $\begin{array}{c}{L}_p[a,b]:=\{f(t):[a,b]\to C|\underset{t=a}{\overset{b}{{\displaystyle \int }}}|f(t){|}^{p}dt<\infty \\ \text{(here the integral is understood in the Lebesgue sense),}\\ \Vert f{\Vert }_p:={(\underset{t=a}{\overset{b}{{\displaystyle \int }}}|f(t){|}^{p}dt)}^{1/p}\}\end{array}$

Remark 18.1

Sure, here functions f(t) are not obligatory continuous.

18.1.6 Lebesgue spaces $L_{\infty }\left[a,b\right]$

It contains all measurable functions from [a, b] to $C$ , namely,

(18.8) $\begin{array}{c}{L}_{\infty }\left[a,b\right]:=\{f\left(t\right):\left[a,b\right]\to C|\\ {\Vert f\Vert }_{\infty }:=\underset{t\in \left[a,b\right]}{ess\text{ sup}}\text{|f}\left(t\right)|<\infty \}\end{array}$

18.1.7 Sobolev spaces $S_p^l\left(G\right)$

It consists of all functions (for simplicity, real valued) f (t) defined on G which have p -integrable continuous derivatives $f^{\left(i\right)}\left(t\right)\text{undefined}\left(i=1,\dots ,l\right)$ , that is,

(18.9) $\begin{array}{c}{S}_p^l[G]:=\{f(t):G\to ℝ|<\infty (i=1,\dots ,l)\\ \text{ (the integral is understood in the Lebesgue sense),}\\ {\Vert f\Vert }_{S_p^l(G)}:={(\underset{t\in G}{{\displaystyle \int }}|f(t){|}^{p}dt+{\displaystyle \sum }_{i=1}^l\underset{t\in G}{{\displaystyle \int }}|f^{(i)}(t){|}^{p}dt)}^{1/p}\}\end{array}$

More exactly, the Sobolev space is the completion (see definition below) of (18.9).

18.1.8 Frequency domain spaces $L_p^{m\times k},R{L}_p^{m\times k},L_{\infty }^{m\times k}and\text{undefined}R{L}_{\infty }^{m\times k}$

By definition

1.

The Lebesgue space $L_p^{m\times k}$ is the space of all p-integrable complex matrices, i.e.,

(18.10) $\begin{array}{l}{L}_p^{m\times k}:=\{F:C\to C^{m\times k}|\\ {\Vert F\Vert }_{L_p^{m\times k}}:={\left(\frac{1}{2\pi }{\displaystyle \underset{\omega =-\infty }{\overset{\infty }{\int }}{\left(tr\left\{F\left(j\omega \right)F^{~}\left(j\omega \right)\right\}\right)}^{P-1}d\omega }\right)}^{1/p}<\infty \}\\ \left(\begin{array}{cc}\text{here}& F^{˜}\left(j\omega \right):=F^{\text{T}}\left(-j\omega \right)\end{array}\right)\end{array}$

2.

The Lebesgue space $R{L}_p^{m\times k}$ is the subspace of $L_p^{m\times k}$ containing only complex matrices with rational elements, i.e., in

$F={\Vert F_{i,j}\left(s\right)\Vert }_{i=\overline{1,m};j=\overline{1,k}}$

each element F_{i, j} (s) represents the polynomial ratio

(18.11) $\begin{array}{l}{F}_{i,j}(s)=\frac{a_{i,j}^0+a_{i,j}^{1}s+\dots +a_{i,j}^{pi,j}{s}_{i,j}^{pi}}{b_{i,j}^0+b_{i,j}^{1}s+\dots +b_{i,j}^{qi,j}{s}_{i,j}^{qi,j}}\\ \begin{array}{cccc}{p}_{i,j}& \text{and}& q_{i,j}& \text{are positive integers}\end{array}\end{array}$

Remark 18.2

if $p_{i,j}\le q_{i,j}$ for each element F_ij of (18.11), then F (s) can be interpreted as a matrix transfer function of a linear (finite-dimensional) system.

3.

The Lebesgue space $L_{\infty }^{m\times k}$ is the space of all complex matrices bounded (almost everywhere) on the imaginary axis elements, i.e.,

(18.12) $\begin{array}{c}{L}_{\infty }^{m\times k}:=\{F:C\to C^{m\times k}|\\ \underset{=\underset{\omega \in \left(-\infty ,\infty \right)}{ess\text{ sup}}{\lambda }_{\max }^{1/2}\left\{F\left(j\omega \right)F^{\sim }\left(j\omega \right)\right\}<\infty }{{\Vert F\Vert }_{L_{\infty }^{m\times k}}:=\underset{s:\mathrm{Re}\text{undefined}s>0}{ess\text{ sup}}{\lambda }_{\max }^{1/2}\left\{F\left(s\right)F^{\sim }\left(s\right)\right\}\}}\end{array}$

(the last equality may be regarded as the generalization of the maximum modulus principle 17.10 for matrix functions).

4.

The Lebesgue space $R{L}_{\infty }^{m\times k}$ is the subspace of $L_{\infty }^{m\times k}$ containing only complex matrices with rational elements given in the form (18.11).

18.1.9 Hardy spaces $H_p^{m\times k},R{H}_p^{m\times k},H_{\infty }^{m\times k}\text{undefined}and\text{undefined}R{H}_{\infty }^{m\times k}$

The Hardy spaces $H_p^{m\times k},R{H}_p^{m\times k},H_{\infty }^{m\times k}\text{undefined}and\text{undefined}R{H}_{\infty }^{m\times k}$ are subspaces of the corresponding Lebesgue spaces $L_p^{m\times k},R{L}_p^{m\times k},L_{\infty }^{m\times k}\text{undefined}and\text{undefined}R{L}_{\infty }^{m\times k}$ containing complex matrices with only regular (holomorphic) (see Definition 17.2) elements on the open half-plane Re s > 0.

Remark 18.3

If p_i,j ≤ q_i,j for each element F_ij of, then $F\left(s\right)\in R{H}_p^{m\times k}$ can be interpreted as a matrix transfer function of a stable linear (finite-dimensional) system.

Example 18.1

$\begin{array}{cc}\frac{1}{2-s}\in R{L}_2:=R{L}_{{}_2}^{1\times 1},& \frac{1-s}{2-s}\in R{L}_{\infty }:=R{L}_{{}_{\infty }}^{1\times 1}\\ \frac{1}{2+s}\in H{L}_2:=H{L}_{{}_2}^{1\times 1},& \frac{1-s}{2+s}\in R{H}_{\infty }:=R{L}_{\infty }^{1\times 1}\\ \frac{e^{-s}}{2-s}\in L_2:=L_2^{1\times 1}& \frac{e^{-s}}{2+s}\in H_2:=H_2^{1\times 1}\\ e^{-1}\frac{1-s}{2-s}\in L_{\infty }:=L_{\infty }^{1\times 1}& e^{-s}\frac{1-s}{2+s}\in H_{\infty }:=H_{\infty }^{1\times 1}\end{array}$

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Fracture Mechanics of Graded Materials

F. Erdogan , in Functionally Graded Materials 1996, 1997

3 MAJOR ISSUES IN FGMS

The principles of fracture mechanics described in the previous section are applicable to inhomogeneous as well as homogeneous materials. In FGMs the difficulties arise in the solution of elastic or elastic–plastic crack problems to evaluate $\mathcal{G}$ or k ₁, k ₂, k ₃ and in characterizing the material to determine K_IC, ${\mathcal{G}}_{IC}$ or ${\mathcal{G}}_c$ where the fracture toughness ${\mathcal{G}}_{IC}$ is no longer a material constant [5]. The definitions of stress intensity factors and expressions of the strain energy release rates given by (4) and (5) are still valid provided the elastic parameters μ and κ are evaluated at the crack tip. Following are some of the major issues concerning the fracture mechanics of FGMs.

(a)

Elastic singularities. As long as the elastic parameters μ and κ are continuous functions of the space variables with piecewise continuous derivatives, the stress state around the crack tips has the standard square-root singularity. For example, in plane isotropic elasticity problems for r  →   0 the leading terms of the stresses are given by [6–8]

(6) ${\sigma }_{ij}\simeq \frac{\varphi \left(r,\theta \right)}{\sqrt{2r}}\left[k_1{f}_{1ij}\left(\theta \right)+k_2{f}_{2ij}\left(\theta \right)\right],\left(i,j\right)=\left(r,\theta \right)$

where (r, θ) are the polar coordinates at the crack tip, k ₁ and k ₂ are the modes I and II stress intensity factors, φ(r, θ) is a smooth function with φ(0, θ) =   1 and the functions f _1ij and f _2ij are identical to that found for crack problems in isotropic homogeneous materials. If the crack tip terminates at a kink or slope discontinuity of μ(x), there would be no change in the dominant terms shown in (6) [8]. However, for small values of r the next significant term would be different, which, for the cleavage stress at the crack tip is given by [9]

(7) ${\sigma }_{\theta \theta }\left(r,0\right)\simeq \frac{k_1}{\sqrt{2r}}\left(1+\frac{1-4v}{8\pi \left(1-v\right)}\beta r\ln r\right)+c\sqrt{r}\text{,}$

where c is a constant, μ(x) =   μ ₀exp(βx) for x <  0, μ   =   μ ₀ for x >   0 and the crack is located along 0   < x   <   a, y =  0.

(b)

Analytical methods/benchmark solutions. Even though there are no known closed form solutions for crack problems in FGMs, for simple property variations the formulations leading to singular integral equations are straightforward and accurate solutions can be obtained [7].

(c)

Computational methods. Finite element method is the major computational tool. However, to improve efficiency and accuracy the development of enriched crack tip and transition elements and ordinary inhomogeneous elements will be needed [10].

(d)

Material orthotropy. In many cases the material orthotropy seems to be the consequence of processing technique. For example, FGMs processed by using a plasma spray technique tend to have a lamellar structure. Flattened splats and relatively weak splat boundaries result in an oriented material with higher stiffness and weaker cleavage planes parallel to the boundary [11]. On the other hand graded materials processed by an electron beam physical vapor deposition technique have usually a columnar structure giving higher stiffness and weak fracture planes in thickness direction [12]. These oriented materials can generally be approximated by an inhomogeneous orthotropic medium.

(e)

Inelastic behavior. Because of the length scales involved in FGM coatings and interfaces, in addition to conventional plasticity, one may have to use a microplasticity approach which accounts for the effect of strain gradients on strain hardening coefficients. The resulting nonlinear elastic–plastic crack problems require a numerical approach with special inhomogeneous elements.

(f)

Rheological effects. Invariably FGMs are used in high temperature environments. As a result the time-temperature effects may not be negligible and the material may have to be modeled as an inhomogeneous viscoelastic or viscoplastic medium.

(g)

Dynamic effects. Generally, high velocities in propagating cracks and high rates of loading (e.g., impact) in stationary cracks would necessitate the consideration of inertia effects in solving the fracture problem. However, even for the uncracked linear elastic inhomogeneous bounded medium, the stress wave phenomenon is not fully understood. The existing solutions are restricted mostly to one dimensional problems in materials with certain simple property gradings.

Material characterization. This is still the most important issue in studying the fracture mechanics FGMs. The knowledge of thermomechanical and fracture mechanics parameters of the material is essential for any realistic predictive reliability study of FGM components.

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Complex Analysis

Jonathan M. Blackledget , in Digital Signal Processing (Second Edition), 2006

1.4 Complex Integration

The integral of a complex function is denoted by

$I={\displaystyle \underset{C}{\int }f\left(z\right)\mathit{dz}}$

where

$f\left(z\right)=u\left(x,y\right)+\mathit{iv}\left(x,y\right),\mathit{dz}=\mathit{dx}+\mathit{idy}$

and C denotes the 'path' of integration in the complex plane (see Figure 1.6). Hence,

Figure 1.6. Integration in the complex plane is taken to be along a path C

$I={\displaystyle \underset{c}{\int }\left(u+\mathit{iv}\right)\left(\mathit{dx}+\mathit{idy}\right)}={\displaystyle \underset{c}{\int }\left(\mathit{udx}-\mathit{vdy}\right)+i{\displaystyle \underset{c}{\int }\left(\mathit{udy}+\mathit{udx}\right)}}$

with the fundamental results

$\begin{array}{l}{\displaystyle \underset{C}{\int }\left[f\left(z\right)+g\left(z\right)\right]}\mathit{dz}={\displaystyle \underset{C}{\int }f\left(z\right)\mathit{dz}}+{\displaystyle \underset{C}{\int }g\left(z\right)}\mathit{dz},\\ {\displaystyle \underset{C}{\int }\mathit{kf}\left(z\right)\mathit{dz}}=k{\displaystyle \underset{C}{\int }f\left(z\right)\mathit{dz}}\end{array}$

and

${\displaystyle \underset{C_1+C_2}{\int }f\left(z\right)\mathit{dz}}={\displaystyle \underset{C_1}{\int }f\left(z\right)\mathit{dz}}+{\displaystyle \underset{C_2}{\int }f\left(z\right)\mathit{dz}}$

as illustrated in Figure 1.7.

Figure 1.7. Integration over two paths C ₁ and C ₂ in the complex plane is given by the sum of the integrals along each path.

Example 1 Integrate f(z) = 1/z from 1 to z along a path C that does not pass through z = 0 (see Figure 1.8), i.e. evaluate the integral

Figure 1.8. Illustration of the path of integration C for integrating 1/z from 1 to z.

$I={\displaystyle \underset{c}{\int }\frac{\mathit{dz}}{z}\text{.}}$

Let z = r exp(iθ), so that dz = dr exp(iθ) + r exp(iθ)idθ. Then

$\begin{array}{l}I={\displaystyle \underset{C}{\int }\frac{\mathit{exp}\left(\mathit{i\theta }\right)\left(\mathit{dr}+\mathit{rid\theta }\right)}{rexp\left(\mathit{i\theta }\right)}={\displaystyle \underset{C}{\int }\left(\frac{\mathit{dr}}{r}+\mathit{id\theta }\right)}}\\ ={\displaystyle \underset{1}{\overset{\left|z\right|}{\int }}\frac{\mathit{dr}}{r}+i}{\displaystyle \underset{0}{\overset{\theta +2\mathit{\pi n}}{\int }}\mathit{d\theta }}\end{array}$

where n is the number of time that the path C encircles the origin in the positive sense, n = 0, ±   1, ±   2, ..... Hence

$I=ln\left|z\right|+i\left(\theta +2\mathit{\pi n}\right)=lnz\text{.}$

Note, that substitutions of the type z = r exp(iθ) are a reoccurring theme in the evaluation and analysis of complex integration.

Example 2 Integrate f(z) =   1/z around z = exp(iθ) where 0 ≤ θ < 2π (see Figure 1.9). Here,

Figure 1.9. Integration of 1/z around z = exp(iθ), 0 ≤ θ ≤ 2π.

$\mathit{dz}=exp\left(\mathit{i\theta }\right)\mathit{id\theta }$

The path of integration C in this case is an example of a contour. A Contour is a simple closed path. The domain or region of the z-plane through which C is chosen must be simply connected (no singularities or other non-differentiable features). Contour integrals are a common feature of complex analysis and will be denoted by ∮ from here on

Important Result

Proof Let z = r exp (iθ), then dz = r exp (iθ)idθ and

$\begin{array}{c}\hfill {\displaystyle \underset{C}{\int }\frac{\mathit{dz}}{z^{n+1}}={\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{rexp\left(\mathit{i\theta }\right)\mathit{id\theta }}{r^{n+1}exp\left[i\left(n+1\right)\theta \right]}=\frac{i}{r^n}{\displaystyle \underset{0}{\overset{2\pi }{\int }}exp\left(-\mathit{in\theta }\right)\mathit{d\theta }.}}}\hfill \\ \hfill =-\frac{i}{r^n}\frac{1}{\mathit{in}}{\left[exp\left(-\mathit{in\theta }\right)\right]}_0^{2\pi }=-\frac{1}{r^n}\frac{1}{n}\left[exp\left(-2\mathit{\pi in}\right)-1\right]=0\text{.}\hfill \end{array}$

Note, that n must be an integer > 0.

1.4.1 Green's Theorem in the Plane

Theorem If S is a closed region in the x – y plane bounded by a simple closed curve C and if P and Q are continuous function of x and y having continuous derivatives in S, then

${\displaystyle \underset{C}{\oint }\left(\mathit{Pdx}+\mathit{Qdy}\right)=}{\displaystyle \underset{S}{\iint }\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)}\mathit{dxdy}$

Proof Consider the accompanying diagram (see Figure 1.10).

Figure 1.10. Path of integration for the proof of Green's theorem in the plane.

Let curve ACB be described by the equation

$y=Y_1\left(x\right)\text{.}$

Let curve BDA be described by the equation

$y=Y_2\left(x\right)\text{.}$

Let curve DAC be described by the equation

$x=X_1\left(y\right)\text{.}$

Let curve CBD be described by the equation

$x=X_2\left(y\right)\text{.}$

Then,

$\begin{array}{c}\hfill {\displaystyle \underset{S}{\iint }\frac{\partial P}{\partial y}\mathit{dxdy}={\displaystyle \underset{a}{\overset{b}{\int }}\left({\displaystyle \underset{Y_1}{\overset{Y_2}{\int }}\frac{\partial P}{\partial y}\mathit{dy}}\right)}}\mathit{dx}={\displaystyle \underset{a}{\overset{b}{\int }}\left[P\left(x,Y_2\right)-P\left(x,Y_1\right)\right]}\mathit{dx}\hfill \\ \hfill ={\displaystyle \underset{a}{\overset{b}{\int }}P\left(x,Y_2\right)\mathit{dx}-{\displaystyle \underset{a}{\overset{b}{\int }}P(x,Y_1})}\mathit{dx}=-{\displaystyle \underset{a}{\overset{b}{\int }}P\left(x,Y_1\right)\mathit{dx}-{\displaystyle \underset{b}{\overset{a}{\int }}P\left(x,Y_2\right)\mathit{dx}=-{\displaystyle \underset{C}{\oint }\mathit{Pdx}\text{.}}}}\hfill \end{array}$

Similarly,

$\begin{array}{c}\hfill {\displaystyle \underset{S}{\iint }\frac{\partial Q}{\partial x}}\mathit{dxdy}={\displaystyle \underset{c}{\overset{d}{\int }}\left({\displaystyle \underset{X_1}{\overset{X_2}{\int }}\frac{\partial Q}{\partial x}\mathit{dx}}\right)}\mathit{dy}={\displaystyle \underset{c}{\overset{d}{\int }}\left[Q\left(X_2,\mathrm{y}\right)-\mathrm{Q}\left(X_1,\mathrm{y}\right)\right]\mathit{dy}}\hfill \\ \hfill ={\displaystyle \underset{c}{\overset{d}{\int }}Q\left(X_2,y\right)}\mathit{dy}-{\displaystyle \underset{c}{\overset{d}{\int }}Q\left(X_1,y\right)\mathit{dy}=}{\displaystyle \underset{c}{\overset{d}{\int }}Q\left(X_2,y\right)}\mathrm{dy}+{\displaystyle \underset{d}{\overset{c}{\int }}Q\left(X_1,y\right)}\mathit{dy}={\displaystyle \underset{C}{\oint }\mathit{Qdy}\text{.}}\hfill \end{array}$

1.4.2 Cauchy's Theorem

Theorem If f(z) is analytic and f′(z) is continuous in a simply connected region R, and C is a simple closed curve lying within R, then

${\displaystyle \underset{C}{\oint }f\left(z\right)\mathit{dz}=0\text{.}}$

Proof

${\displaystyle \underset{C}{\oint }f\left(z\right)\mathit{dz}}={\displaystyle \underset{C}{\oint }\left(\mathit{udx}-\mathit{vdy}\right)}+i{\displaystyle \underset{C}{\oint }\left(\mathit{udy}+\mathit{vdx}\right)}\text{.}$

Using Green's theorem in the plane, i.e.

${\displaystyle \underset{C}{\oint }\left(\mathit{Pdx}+\mathit{Qdy}\right)={\displaystyle \underset{S}{\iint }\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)}\mathit{dxdy}}$

where P and Q have continuous partial derivatives, we get

${\displaystyle \underset{C}{\oint }f\left(z\right)\mathit{dz}={\displaystyle \underset{S}{\iint }\left(-\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)}}\mathit{dxdy}+i{\displaystyle \underset{S}{\iint }\left(\frac{\partial v}{\partial y}-\frac{\partial u}{\partial x}\right)}\mathit{dxdy}\text{.}$

But from the Cauchy-Riemann equations

$\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$

Corollary 1 If C ₁ and C ₂ are two paths joining points a and z in the z-plane (see Figure 1.11) then, provided f(z) is analytic at all points on C ₁ and C ₂ and between C ₁ and C ₂,

Figure 1.11. An integral is independent of the path that is taken in the complex plane.

${\displaystyle \underset{a|C_1\uparrow }{\overset{z}{\int }}f\left(z\right)}\mathit{dz}={\displaystyle \underset{a|C_2\uparrow }{\overset{z}{\int }}f\left(z\right)\mathit{dz}\text{.}}$

Here, ↑ denotes that the path of integration is in an anti-clockwize direction and ↓ is taken to denote that the path of integration is in a clockwize direction. This result comes from the fact that the path taken is independent of the integral since from Cauchy's theorem

$\begin{array}{l}\left({\displaystyle \underset{a|C_1\uparrow }{\overset{Z}{\int }}+}{\displaystyle \underset{z|C_{2\downarrow }}{\overset{a}{\int }}}\right)f\left(z\right)\mathit{dz}=0\\ \end{array}$

and thus,

${\displaystyle \underset{a|C_1\uparrow }{\overset{z}{\int }}f\left(z\right)}\mathit{dz}=-{\displaystyle \underset{z|C_2\downarrow }{\overset{a}{\int }}f\left(z\right)\mathit{dz}}={\displaystyle \underset{a|C_2\uparrow }{\overset{z}{\int }}f\left(z\right)}\mathit{dz}\text{.}$

Corollary 2 If f(z) has no singularities in the annular region between the contours C ₁ and C ₂, then

${\displaystyle \underset{C_1\uparrow }{\oint }f\left(z\right)\mathit{dz}}={\displaystyle \underset{C_2\uparrow }{\oint }f\left(z\right)\mathit{dz}}\text{.}$

We can show this result by inserting a cross-cut between C ₁ and C ₂ to produce a single contour at every point within which f(z) is analytic (Figure 1.12). Then, from Cauchy's theorem

Figure 1.12. Two contours C ₁ and C ₂ made continuous through a cross-cut.

${\displaystyle \underset{C_1\uparrow }{\int }f\left(z\right)\mathit{dz}}+{\displaystyle \underset{a}{\overset{b}{\int }}f\left(z\right)\mathit{dz}-{\displaystyle \underset{C_2\downarrow }{\int }f\left(z\right)\mathit{dz}}+{\displaystyle \underset{c}{\overset{d}{\int }}f\left(z\right)\mathit{dz}}=0\text{.}}$

Rearranging,

${\displaystyle \underset{C_1\uparrow }{\int }f\left(z\right)}\mathit{dz}+{\displaystyle \underset{a}{\overset{b}{\int }}f\left(z\right)}\mathit{dz}-{\displaystyle \underset{d}{\overset{c}{\int }}f\left(z\right)\mathit{dz}={\displaystyle \underset{C_2\uparrow }{\int }f\left(z\right)\mathit{dz}\text{.}}}$

and hence,

${\displaystyle \underset{C_1\uparrow }{\oint }f\left(z\right)\mathit{dz}}={\displaystyle \underset{C_2\uparrow }{\oint }f\left(z\right)\mathit{dz}\text{.}}$

1.4.3 Defining a Contour

A contour can be defined around any number of points in the complex plane. Thus, if we consider three points z ₁, z ₂ and z ₃, for example, then the paths Г₁, Г₂ and Г₃ respectively can be considered to be simply connected as shown in Figure 1.13. Thus, we have

Figure 1.13. Defining the contour C which encloses three points in the complex plane.

${\displaystyle \underset{C}{\oint }f\left(z\right)\mathit{dz}={\displaystyle \underset{{\text{Γ}}_1}{\int }f\left(z\right)\mathit{dz}+}}{\displaystyle \underset{\text{Γ}2}{\int }f\left(z\right)\mathit{dz}+}{\displaystyle \underset{\text{Γ}3}{\int }f\left(z\right)\mathit{dz}\text{.}}$

where Γ is a circle which can be represented in the form z = r exp(iθ), 0 ≤ θ ≤ 2π. Then, dz = r exp(iθ)idθ and

${\displaystyle \underset{\text{Γ}}{\int }\frac{\mathit{dz}}{z}={\displaystyle \underset{0}{\overset{2\pi }{\int }}\mathit{id\theta }=2\mathit{\pi i}\text{.}}}$

1.4.4 Example of the Application of Cauchy's Theorem

Let us consider the evaluation of the integral

${\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{sinx}{x}\mathit{dx}\text{.}}$

Consider the complex function exp(iz)/z which has one singularity at z = 0. Note that we have chosen the function exp(iz)/z because it is analytic everywhere on and within the contour illustrated in Figure 1.14. By Cauchy's theorem,

Figure 1.14. Contour used for evaluating the integral of sin(x)/x, x ∈ [0,∞).

${\displaystyle \underset{C_R}{\int }\frac{exp\left(\mathit{iz}\right)}{z}\mathit{dz}+{\displaystyle \underset{-R}{\overset{-r}{\int }}\frac{exp\left(\mathit{ix}\right)}{x}\mathit{dx}+{\displaystyle \underset{C_r}{\int }\frac{exp\left(\mathit{iz}\right)}{z}\mathit{dz}+{\displaystyle \underset{r}{\overset{R}{\int }}\frac{exp\left(\mathit{ix}\right)}{x}\mathit{dx}}=0\text{.}}}}$

We now evaluate the integrals along the real axis:

$\begin{array}{c}\hfill {\displaystyle \underset{r}{\overset{R}{\int }}\frac{exp\left(\mathit{ix}\right)}{x}\mathit{dx}+{\displaystyle \underset{-R}{\overset{-r}{\int }}\frac{exp\left(\mathit{ix}\right)}{x}\mathit{dx}={\displaystyle \underset{r}{\overset{R}{\int }}\frac{exp\left(\mathit{ix}\right)}{x}\mathit{dx}-{\displaystyle \underset{r}{\overset{R}{\int }}\frac{exp\left(-\mathit{ix}\right)}{x}}\mathit{dx}={\displaystyle \underset{r}{\overset{R}{\int }}\frac{exp\left(-\mathit{ix}\right)-exp\left(-\mathit{ix}\right)}{x}\mathit{dx}}}}}\hfill \\ \hfill =2i{\displaystyle \underset{r}{\overset{R}{\int }}\frac{sinx}{x}\mathit{dx}}=2i{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{sinx}{x}\mathit{dx}}\mathrm{as}R\to \infty \mathrm{and}r\to 0\text{.}\hfill \end{array}$

Evaluating the integral along C_r :

$\begin{array}{c}\hfill {\displaystyle \underset{C_r}{\int }\frac{exp\left(\mathit{iz}\right)}{z}\mathit{dz}}={\displaystyle \underset{\pi }{\overset{0}{\int }}exp\left[\mathit{ir}\left(cos\theta +isin\theta \right)\right]}\mathit{id\theta }\left[\mathrm{with}z=rexp\left(\mathit{i\theta }\right)\right]\hfill \\ \hfill =-{\displaystyle \underset{0}{\overset{\pi }{\int }}exp\left[\mathit{ir}\left(cos\theta +isin\theta \right)\mathit{id\theta }=-{\displaystyle \underset{0}{\overset{\pi }{\int }}exp\left(0\right)}\mathit{id\theta }\mathrm{as}r\to 0\left(\mathrm{i}.\mathrm{e}.‐\mathit{i\pi }\right)\right]}\text{.}\hfill \end{array}$

Evaluating the integral along C_R :

$\begin{array}{c}\hfill {\displaystyle \underset{C_R}{\int }\frac{exp\left(\mathit{iz}\right)}{z}}\mathit{dz}={\displaystyle \underset{0}{\overset{\pi }{\int }}exp[\mathit{iR}\left(cos\theta +isin\theta \right)\mathit{id\theta }\left[\mathrm{with}z=Rexp\left(\mathit{i\theta }\right)\right]}\hfill \\ \hfill ={\displaystyle \underset{0}{\overset{\pi }{\int }}exp\left(\mathit{iR}cos\theta \right)exp\left(-Rsin\theta \right)\mathit{id\theta }=0\mathrm{as}R\to \infty \text{.}}\hfill \end{array}$

Combining the results,

$2i{\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{sinx}{x}\mathit{dx}}-\mathit{i\pi }=0\text{.}$

1.4.5 Cauchy's Integral Formula

Theorem If f(z) is analytic in a simply connected region R and C is a contour that lies within R and encloses point z₀, then

${\displaystyle \underset{C}{\oint }\frac{f\left(z\right)}{z-z_0}\mathit{dz}}=2\mathit{\pi if}\left(z_0\right)\text{.}$

Proof Consider a small circle Γ with center z ₀ and radius r, lying entirely within C (see Figure 1.15).

Figure 1.15. Circular contour Γ within arbitrary contour C enclosing a point z ₀.

Then

$I={\displaystyle \underset{C}{\oint }\frac{f\left(z\right)}{z-z_0}\mathit{dz}}={\displaystyle \underset{\text{Γ}}{\oint }\frac{f\left(z\right)}{z-z_0}\mathit{dz}\text{.}}$

Let z – z ₀ = r exp(iθ), then dz = r exp(iθ)idθ and

$\begin{array}{c}\hfill I={\displaystyle \underset{0}{\overset{2\mathrm{\pi }}{\int }}\frac{f\left[z_0+rexp\left(\mathit{i\theta }\right)\right]}{rexp\left(\mathit{i\theta }\right)}rexp\left(\mathit{i\theta }\right)}\mathit{id\theta }=i{\displaystyle \underset{0}{\overset{2\mathrm{\pi }}{\int }}f\left[z_0+rexp\left(\mathit{i\theta }\right)\right]}\mathit{d\theta }=i{\displaystyle \underset{0}{\overset{2\mathrm{\pi }}{\int }}f\left(z_0\right)}\mathit{d\theta }\mathrm{as}r\to 0\hfill \\ \hfill =2\mathrm{\pi }if\left(z_0\right)\text{.}\hfill \end{array}$

Hence,

$f\left(z_0\right)=\frac{1}{2\mathit{\pi i}}{\displaystyle \underset{C}{\oint }\frac{f\left(z\right)}{z-z_0}\mathit{dz}\text{.}}$

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Variational Calculus and Optimal Control

Alexander S. Poznyak , in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, Volume 1, 2008

Proof

For a constant c, the function $\upsilon \left(x\right):={\displaystyle {\int }_{t=a}^x\left[h\left(t\right)-c\right]dt}$ is in $C^1\left[a,b\right]$ (it has a continuous derivative) and ${\upsilon }^{\text{'}}\left(x\right)=h\left(x\right)-c$ so that $\upsilon \left(a\right)=0$ . It will be in $D_1$ if, additionally, it satisfies the condition $\upsilon \left(b\right)=0$ , that is, if $\upsilon \left(b\right):={\displaystyle {\int }_{t=a}^b\left[h\left(t\right)-c\right]dt=0}$ , or $c={\left(b-a\right)}^{-1}{\displaystyle {\int }_{t=a}^{b}h\left(t\right)dt}$ . Thus, for these c and $\upsilon \left(x\right)$ , in view of (22.1), we have

$\begin{array}{ll}0\le {\displaystyle \underset{x=a}{\overset{b}{\int }}{\left[h\left(x\right)-c\right]}^{2}dx}\hfill & ={\displaystyle \underset{x=a}{\overset{b}{\int }}\left[h\left(x\right)-c\right]{\upsilon }^{\text{'}}\left(x\right)dx}\hfill \\ \hfill & ={\displaystyle \underset{x=a}{\overset{b}{\int }}h\left(x\right){\upsilon }^{\text{'}}\left(x\right)dx-c\upsilon \left(x\right)|{}_{x=a}^{x=b}=0}\hfill \end{array}$

and, by Lemma 22.1, it follows that ${\left[h\left(x\right)-c\right]}^2\equiv 0$ which completes the proof.

The next lemma generalizes Lemma 22.2.

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