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A meromorphic function is analytic everywhere in the complex plane except at a nite number of poles EXAMPLE 513 3 Describe the singularities of f ( z ) = 1/[( z 2)( z + 4) ] Is this function entire SOLUTION The function f ( z ) has singularities at z = 2 and z = 4 The pole at z = 2 is a simple pole because the power of this term is 1 The pole at z = 4 is a pole of order 3 The function is not entire, because it is not analytic at the poles Since there are a nite number of poles, the function is meromorphic EXAMPLE 514 Suppose that f ( z ) = ( z 1)cos[1/ (z + 2)] Find the Laurent expansion of this function about the point z = 2 and describe the nature of any singularities Identify the analytic and principal parts of the series expansion SOLUTION Recall the power series expansion of the cosine function: 1 1 1 cos z = 1 z 2 + z 4 z 6 + 2 4! 6!

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The Cisco 2500 Series provides router functionality to networks that only require data transport (voice is not supported) There are several types of routers in this series, including LAN routers, router/hub combinations, access servers, high-density serial routers, dual LAN routers, and single LAN routers This series is the most commonly used to connect branch offices to corporate headquarters It supports Ethernet, Token Ring, and ISDN Figure 4-5 illustrates an environment using several models of the 2500 Series

Now let w = z + 2 to simplify notation Then 1 1 1 1 1 1 1 1 cos = cos = 1 + + w z + 2 2 w 4! w 6! w The term z 1 = w 3 and so 1 1 f ( z ) = ( z 1) cos = ( w 3) cos w z + 2 1 1 2 1 1 4 1 1 6 = ( w 3) 1 + + 2 w 4! w 6! w

PVC (Polyvinyl Chloride) A material used for the outside insulation and jacketing of most cables Also a fire rating for a type of cable that has no significant fire protection Quality of Service (QoS) Policies that control how much bandwidth a protocol, PC, user, VLAN, or IP address may use Raceway Cable organizing device that adheres to walls, making for a much simpler, though less neat, installation than running cables in the walls RADIUS Server A system that enables remote users to connect to a network service RAID (Redundant Array of Independent [or Inexpensive] Devices [or Disks]) A way of creating a fault-tolerant storage system There are six levels Level 0 uses byte-level striping and provides no fault tolerance Level 1 uses mirroring or duplexing Level 2 uses bit-level striping Level 3 stores error-correcting information (such as parity) on a separate disk, and uses data striping on the remaining drives Level 4 is level 3 with block-level striping Level 5 uses block-level and parity data striping RCP (Remote Copy) Provides the capability to copy files to and from the remote server without the need to resort to FTP or NFS (Network File System, UNIX form of folder sharing) RCP can also be used in scripts and shares TCP port 514 with RSH Real-Time Processing The processing of transactions as they occur, rather than batching them Pertaining to an application, processing in which response to input is fast enough to affect subsequent inputs and guide the process and in which records are updated immediately The lag from input time to output time must be sufficiently small for acceptable timeliness Timeliness is a function of the total system: missile guidance requires output within a few milliseconds of input, whereas scheduling of steamships requires a response time in days Real-time systems are those with a response time of milliseconds; interactive systems respond in seconds; and batch systems may respond in hours or days.

The Cisco 2600 Series gives remote branch offices a solution for data/voice/video integration, departmental dial services, and VPN access These routers provide some versatility with their capability of sharing modular interfaces with the Cisco 1600 and 3600 Series Figure 4-6 illustrates an environment using the 2600 Series