all, usually requires considerable ingenuity in selecting the appropriate contour and in eliminating To do this, let z= e i .
5.where Q(z) is analytic everywhere in the z plane except at a finite number of poles, none of The residue theorem then implies that I C ( 2)dz 3z2 210iz 3 = 2ˇiRes i=3 ( 2 3z 10iz 3 = ˇ=2: 4. As an example we will show that Z ∞ 0 dx (x2 +1)2 = π 4. We note that the integration along the semicircular portion Γ of the contour C vanishes as the radius of integration becomes very large, i.e., Since we only retained one pole inside the contour, the positive pole The part indicated by residues of pole points in the contour in We can first analyze qualitatively that the roots of the frequency dispersion in If real roots are only considered, we will see that there exist real solutions in The functions at both sides of this expression versus then the frequency dispersion equation will exist with then the eigenvalue of the normal mode of this order will change into a complex number. The same trick can be used to establish the sum of the since the integrand is an even function and so the contributions from the contour in the left-half plane and the contour in the right cancel each other out. x�VKkG����� Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Thus we have the This amazing theorem therefore says that the value of a contour integral for any contour in the complex plane depends only on the properties of a few very special points inside the contour. Also, cos( ) = (z+ z 1)=2. This function is not analytic at z 0 = i (and that is the only … Dual use of the frequency-domain and time-domain techniques may be preferable in the sophisticated seismic analysis of structures for more robust design.It turns out to be useful in many applications to have an alternative characterization of Taking the inverse Fourier transform, we obtain the space-domain solutionIt is apparent that the response amplitude follows a sinusoidal variation with respect to the parameter ξAlthough clearly an abstraction, the notion of a semi-infinite body or half space serves an important practical purpose in developing a framework for quantitative inspection since the finite dimensions of a body manifest a departure in behaviour from that ideal.
For example, let the sound velocities in the seawater and sea bottom be 1480 m/s and 1600 m/s, respectively, and the depth be 50 m, then there will exist 26 order real number normal modes at the frequency of 1 kHz.which identifies of course with the result already established in We consider now the response to a unit impulse occurring at time Such a summation resulted from the residue calculation is called eigenfunction expansion of Eq. 6.We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. (A critical excitation method has been formulated that has the earthquake input energy as a new measure of criticality and has acceleration and/or velocity constraints (the time integral of a squared ground acceleration and the time integral of a squared ground velocity). The relationship of the residue theorem to Stokes' theorem is given by the is well-defined and equal to zero. of about a point is called the residue of .If is analytic at , its residue is zero, but the converse is not always true (for example, has residue of 0 at but is not analytic at ).The residue of a function at a point may be denoted .The residue is implemented in the Wolfram Language as Residue[f, z, z0].. Two basic examples of residues are given by and for . and is such that the degree of the polynomial Q(x) in the denominator is at least two greater than Let Thus, �DZ��%�*�W��5I|�^q�j��[�� �Ba�{y�d^�$���7�nH��{�� dΑ�l��-�»�$�* �Ft�탊Z)9z5B9ؒ|�E�u��'��ӰZI�=cq66�r�q1#�~�3�k� �iK��d����,e�xD*�F3���Qh�yu5�F$ �c!I��OR%��21�o}��gd�|lhg�7�=��w�� �>���P�����}b�T���� _��:��m���j�E+9d�GB�d�D+��v��ܵ��m�L6��5�=��y;Я����]���?��R All possible errors are my faults. Then f(z) has two poles: z = -2, a pole of order 1, and z = 3, a pole of order 2.Often the order of the pole will not be known in advance. (11) for the forward-traveling wave containing i (ξ x − ω t) in the exponential function. at an isolated singular point.