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<626567696E7320> Tj 1 0 0 1 209.04 669.60 Tm <7769746820> Tj 1 0 0 1 234.00 669.60 Tm <646566696E696E6720> Tj 1 0 0 1 276.48 669.36 Tm <6920> Tj 1 0 0 1 282.72 672.24 Tm <3A20> Tj 1 0 0 1 294.48 668.64 Tm <4A2D20> Tj 1 0 0 1 312.96 669.36 Tm <4920> Tj 1 0 0 1 323.28 669.36 Tm <726F20> Tj 1 0 0 1 337.92 669.12 Tm <7468617420> Tj 1 0 0 1 358.08 669.36 Tm <69323A20> Tj 1 0 0 1 378.00 671.52 Tm <2D312E20> Tj 1 0 0 1 394.08 669.36 Tm <496E20> Tj 1 0 0 1 407.52 669.60 Tm <67656E6572616C20> Tj 1 0 0 1 447.12 669.60 Tm <2862757420> Tj 1 0 0 1 468.96 669.12 Tm <6E6F74> Tj 0.24 Tc 1 0 0 1 89.76 655.68 Tm <616C77617973292C20> Tj 1 0 0 1 133.68 655.68 Tm <636F6D706C657820> Tj 1 0 0 1 177.12 655.68 Tm <6E2D756D6265727320> Tj 1 0 0 1 222.00 655.44 Tm <6861766520> Tj 1 0 0 1 247.92 655.44 Tm <6120> Tj 1 0 0 1 255.84 655.44 Tm <7265616C20> Tj 1 0 0 1 276.48 655.68 Tm <7061727420> Tj 1 0 0 1 298.80 655.20 Tm <616E6420> Tj 1 0 0 1 319.44 655.20 Tm <616E20> Tj 1 0 0 1 333.84 655.20 Tm <696D6167696E61727920> Tj 1 0 0 1 384.96 655.68 Tm <706172742E20> Tj 1 0 0 1 410.64 655.20 Tm <5468657920> Tj 1 0 0 1 439.44 655.20 Tm <61726520> Tj 1 0 0 1 456.96 655.20 Tm <7772697474656E20> Tj 1 0 0 1 493.92 654.96 Tm <696E> Tj 0.24 Tc 1 0 0 1 89.76 641.76 Tm <7365766572616C20> Tj 1 0 0 1 127.20 641.76 Tm <666F726D732E20> Tj 1 0 0 1 160.80 641.76 Tm <74686520> Tj 1 0 0 1 178.08 641.76 Tm <6D6F737420> Tj 1 0 0 1 204.72 641.76 Tm <636F6D6D6F6E20> Tj 1 0 0 1 249.84 641.52 Tm <6265696E65> Tj 1.50 Tc 1 0 0 1 282.96 627.60 Tm <7A3A20> Tj 1 0 0 1 301.44 627.60 Tm <612A6269> Tj 0.24 Tc 1 0 0 1 89.04 614.40 Tm <776865726520> Tj 1 0 0 1 121.92 614.16 Tm <7A20> Tj 1 0 0 1 130.80 614.16 Tm <697320> Tj 1 0 0 1 141.60 614.16 Tm <74686520> Tj 1 0 0 1 158.88 614.40 Tm <77686F6C6520> Tj 1 0 0 1 191.04 614.16 Tm <636F6D706C657820> Tj 1 0 0 1 235.68 613.92 Tm <6E756D6265722C20> Tj 1 0 0 1 278.64 613.92 Tm <6120> Tj 1 0 0 1 286.80 613.92 Tm <697320> Tj 1 0 0 1 297.60 613.68 Tm <74686520> Tj 1 0 0 1 315.36 613.68 Tm <7265616C20> Tj 1 0 0 1 336.48 613.92 Tm <706172742C20> Tj 1 0 0 1 361.68 613.68 Tm <616E6420> Tj 1 0 0 1 381.36 613.68 Tm <626920> Tj 1 0 0 1 394.32 613.68 Tm <697320> Tj 1 0 0 1 405.36 613.68 Tm <74686520> Tj 1 0 0 1 423.60 613.68 Tm <696D6167696E61727920> Tj 1 0 0 1 475.68 613.92 Tm <706172742E> Tj 0.48 Tc 1 0 0 1 89.28 585.84 Tm <4F7065726174696F6E7320> Tj 1 0 0 1 150.48 585.84 Tm <6F6E20> Tj 1 0 0 1 165.84 585.84 Tm <436F6D706C657820> Tj 1 0 0 1 214.56 585.84 Tm <4E756D62657273> Tj 0.24 Tc 1 0 0 1 88.32 558.48 Tm <4164646974696F6E20> Tj 1 0 0 1 134.64 558.24 Tm <2620> Tj 1 0 0 1 146.16 558.24 Tm <5375627472616374696F6E3A20> Tj 1 0 0 1 212.16 558.24 Tm <54776F20> Tj 1 0 0 1 237.36 558.24 Tm <636F6D706C657820> Tj 1 0 0 1 281.28 558.00 Tm <6E756D626572732C20> Tj 1 0 0 1 329.28 558.00 Tm <6120> Tj 1 0 0 1 337.44 558.96 Tm <2B20> Tj 1 0 0 1 346.80 557.76 Tm <626920> Tj 1 0 0 1 359.76 557.76 Tm <616E6420> Tj 1 0 0 1 379.92 557.76 Tm <6320> Tj 1 0 0 1 388.08 558.96 Tm <2B20> Tj 1 0 0 1 398.16 557.76 Tm <64692C20> Tj 1 0 0 1 414.24 557.76 Tm <61726520> Tj 1 0 0 1 432.48 557.76 Tm <616464656420> Tj 1 0 0 1 464.40 557.76 Tm <616E64> Tj 0.24 Tc 1 0 0 1 89.04 544.56 Tm <7375627472616374656420> Tj 1 0 0 1 141.84 544.56 Tm <627920> Tj 1 0 0 1 156.96 544.56 Tm <73696D706C7920> Tj 1 0 0 1 192.24 544.32 Tm <636F6D62696E696E6720> Tj 1 0 0 1 246.72 544.32 Tm <74686520> Tj 1 0 0 1 264.24 544.32 Tm <7265616C20> Tj 1 0 0 1 284.88 544.56 Tm <706172747320> Tj 1 0 0 1 312.00 544.08 Tm <616E6420> Tj 1 0 0 1 332.40 544.08 Tm <696D6167696E61727920> Tj 1 0 0 1 383.52 544.32 Tm <706172747320> Tj 1 0 0 1 411.12 543.84 Tm <73657061726174656C792C20> Tj 1 0 0 1 466.80 543.84 Tm <617320> Tj 1 0 0 1 479.76 543.84 Tm <666F6C6C6F77733A> Tj 0.72 Tc 1 0 0 1 197.28 517.44 Tm <28612B20> Tj 1 0 0 1 219.12 516.72 Tm <62692920> Tj 1 0 0 1 235.68 517.68 Tm <2B20> Tj 1 0 0 1 245.76 517.20 Tm <286320> Tj 1 0 0 1 258.00 517.68 Tm <2B20> Tj 1 0 0 1 267.60 516.48 Tm <64692920> Tj 1 0 0 1 283.68 519.36 Tm <3A20> Tj 1 0 0 1 293.76 516.96 Tm <286120> Tj 1 0 0 1 306.00 517.44 Tm <2B20> Tj 1 0 0 1 316.08 516.48 Tm <632920> Tj 1 0 0 1 328.32 517.44 Tm <2B20> Tj 1 0 0 1 338.16 516.96 Tm <286220> Tj 1 0 0 1 351.12 517.44 Tm <2B20> Tj 1 0 0 1 361.20 516.24 Tm <642969> Tj 0.48 Tc 1 0 0 1 197.28 503.28 Tm <286120> Tj 1 0 0 1 209.28 503.76 Tm <2B20> Tj 1 0 0 1 218.88 502.56 Tm <62692920> Tj 1 0 0 1 235.20 505.20 Tm <2D20> Tj 1 0 0 1 245.04 503.04 Tm <286320> Tj 1 0 0 1 256.80 503.52 Tm <2B20> Tj 1 0 0 1 266.88 502.32 Tm <64692920> Tj 1 0 0 1 282.96 505.20 Tm <3A20> Tj 1 0 0 1 293.04 502.80 Tm <286120> Tj 1 0 0 1 305.28 504.72 Tm <2D20> Tj 1 0 0 1 312.24 502.32 Tm <632920> Tj 1 0 0 1 324.72 503.28 Tm <2B20> Tj 1 0 0 1 334.80 499.92 Tm <4F20> Tj 1 0 0 1 347.04 504.96 Tm <2D20> Tj 1 0 0 1 356.64 502.32 Tm <642969> Tj 0.48 Tc 1 0 0 1 88.08 474.72 Tm <4D756C7469706C69636174696F6E3A20> Tj 1 0 0 1 168.00 474.48 Tm <54776F20> Tj 1 0 0 1 192.48 474.48 Tm <636F6D706C657820> Tj 1 0 0 1 237.12 474.48 Tm <6E756D6265727320> Tj 1 0 0 1 281.52 474.24 Tm <61726520> Tj 1 0 0 1 298.80 474.24 Tm <6D756C7469706C69656420> Tj 1 0 0 1 351.84 474.24 Tm <7573696E6720> Tj 1 0 0 1 380.88 474.24 Tm <6120> Tj 1 0 0 1 388.80 474.48 Tm <70726F6365737320> Tj 1 0 0 1 427.92 475.44 Tm <6A75737420> Tj 1 0 0 1 450.24 474.00 Tm <6C696B6520> Tj 1 0 0 1 470.88 474.00 Tm <462E4F2E4C4C2E> Tj 0.48 Tc 1 0 0 1 194.40 447.36 Tm <28612B20> Tj 1 0 0 1 216.00 446.64 Tm <62692920> Tj 1 0 0 1 233.28 450.24 Tm <2A20> Tj 1 0 0 1 242.16 447.12 Tm <286320> Tj 1 0 0 1 254.16 447.60 Tm <2B20> Tj 1 0 0 1 264.00 446.64 Tm <64692920> Tj 1 0 0 1 280.08 449.28 Tm <3A20> Tj 1 0 0 1 290.16 446.64 Tm <616320> Tj 1 0 0 1 303.60 447.60 Tm <2B20> Tj 1 0 0 1 313.68 446.64 Tm <61646920> Tj 1 0 0 1 331.20 447.36 Tm <2B20> Tj 1 0 0 1 340.80 446.40 Tm <62636920> Tj 1 0 0 1 359.04 447.36 Tm <2B20> Tj 1 0 0 1 368.64 446.40 Tm <6264693220> Tj 1 0 0 1 391.20 449.28 Tm <3A20> Tj 1 0 0 1 401.28 446.88 Tm <28616320> Tj 1 0 0 1 418.80 449.04 Tm <2D62642920> Tj 1 0 0 1 448.08 447.36 Tm <2B20> Tj 1 0 0 1 457.92 446.88 Tm <28616420> Tj 1 0 0 1 475.92 447.12 Tm <2B20> Tj 1 0 0 1 485.52 446.16 Tm <62632969> Tj 0.24 Tc 1 0 0 1 87.84 419.52 Tm <4469766973696F6E3A20> Tj 1 0 0 1 140.16 419.52 Tm <546F20> Tj 1 0 0 1 156.00 419.52 Tm <64697669646520> Tj 1 0 0 1 188.64 419.52 Tm <636F6D706C657820> Tj 1 0 0 1 233.28 419.28 Tm <6E756D626572732C20> Tj 1 0 0 1 280.32 419.28 Tm <776520> Tj 1 0 0 1 297.12 419.04 Tm <6E65656420> Tj 1 0 0 1 323.28 419.04 Tm <746F20> Tj 1 0 0 1 336.00 419.04 Tm <666972737420> Tj 1 0 0 1 358.80 419.04 Tm <6C6561726E20> Tj 1 0 0 1 385.20 418.80 Tm <636F6D706C657820> Tj 1 0 0 1 429.60 418.80 Tm <636F6E6A7567616C65732E20> Tj 1 0 0 1 487.68 418.56 Tm <54776F> Tj 0.24 Tc 1 0 0 1 88.56 405.60 Tm <636F6D706C657820> Tj 1 0 0 1 133.44 405.60 Tm <6E756D6265727320> Tj 1 0 0 1 177.36 405.60 Tm <6172652763616C6C656420> Tj 1 0 0 1 226.80 405.36 Tm <636F6D706C657820> Tj 1 0 0 1 271.20 405.12 Tm <636F6E6A75676174657320> Tj 1 0 0 1 325.68 404.88 Tm <696620> Tj 1 0 0 1 336.00 404.88 Tm <7468657920> Tj 1 0 0 1 359.52 404.88 Tm <6861766520> Tj 1 0 0 1 385.20 404.88 Tm <74686520> Tj 1 0 0 1 403.44 404.88 Tm <73616D6520> Tj 1 0 0 1 431.76 404.88 Tm <7265616C20> Tj 1 0 0 1 453.12 405.36 Tm <7061727420> Tj 1 0 0 1 475.20 404.88 Tm <616E6420> Tj 1 0 0 1 494.88 404.64 Tm <746865> Tj 0.24 Tc 1 0 0 1 88.56 391.68 Tm <737A746D6520> Tj 1 0 0 1 116.40 391.68 Tm <696D6167696E617279706172742C20> Tj 1 0 0 1 192.24 391.44 Tm <7769746820> Tj 1 0 0 1 217.20 391.44 Tm <6F6E6C7920> Tj 1 0 0 1 241.44 391.20 Tm <6F6E6520> Tj 1 0 0 1 261.84 391.20 Tm <6F6620> Tj 1 0 0 1 274.56 391.20 Tm <74686520> Tj 1 0 0 1 292.56 391.20 Tm <7369676E7320> Tj 1 0 0 1 320.40 390.96 Tm <6368616E6765642E20> Tj 1 0 0 1 366.72 391.68 Tm <28497420> Tj 1 0 0 1 380.64 390.96 Tm <697320> Tj 1 0 0 1 392.16 390.96 Tm <66617220> Tj 1 0 0 1 408.48 390.96 Tm <6D6F726520> Tj 1 0 0 1 437.28 390.96 Tm <636F6D6D6F6E20> Tj 1 0 0 1 482.16 390.96 Tm <746F> Tj 0.24 Tc 1 0 0 1 88.32 377.52 Tm <6368616E676520> Tj 1 0 0 1 125.76 377.52 Tm <74686520> Tj 1 0 0 1 143.76 377.52 Tm <7369676E20> Tj 1 0 0 1 166.32 377.52 Tm <6F6620> Tj 1 0 0 1 178.80 377.28 Tm <74686520> Tj 1 0 0 1 196.80 377.28 Tm <696D6167696E61727920> Tj 1 0 0 1 248.40 377.52 Tm <706172742C20> Tj 1 0 0 1 272.88 377.04 Tm <62757420> Tj 1 0 0 1 291.84 377.04 Tm <65697468657220> Tj 1 0 0 1 321.84 377.04 Tm <77696C6C20> Tj 1 0 0 1 343.92 377.04 Tm <776F726B292E20> Tj 1 0 0 1 378.72 376.80 Tm <546875732C20> Tj 1 0 0 1 410.40 376.80 Tm <4920> Tj 1 0 0 1 418.56 377.76 Tm <2B20> Tj 1 0 0 1 428.64 376.80 Tm <6920> Tj 1 0 0 1 435.36 376.80 Tm <616E6420> Tj 1 0 0 1 456.96 376.80 Tm <4920> Tj 1 0 0 1 464.88 379.20 Tm <2D20> Tj 1 0 0 1 471.84 376.80 Tm <6920> Tj 1 0 0 1 481.20 376.80 Tm <617265> Tj 0.24 Tc 1 0 0 1 88.08 363.60 Tm <636F6D706C657820> Tj 1 0 0 1 133.68 363.60 Tm <636F6E6A7567617465732C20> Tj 1 0 0 1 190.08 363.36 Tm <616E6420> Tj 1 0 0 1 210.72 363.36 Tm <736F20> Tj 1 0 0 1 224.40 363.36 Tm <61726520> Tj 1 0 0 1 243.12 363.36 Tm <3120> Tj 1 0 0 1 250.80 364.32 Tm <2B20> Tj 1 0 0 1 260.64 363.36 Tm <6920> Tj 1 0 0 1 267.12 363.36 Tm <616E6420> Tj 1 0 0 1 287.28 365.52 Tm <2D3120> Tj 1 0 0 1 300.00 364.08 Tm <2B20> Tj 1 0 0 1 310.08 363.12 Tm <692E20> Tj 1 0 0 1 319.20 362.88 Tm <4E6F746520> Tj 1 0 0 1 346.08 363.12 Tm <7468617420> Tj 1 0 0 1 367.20 363.12 Tm <616C74686F75676820> Tj 1 0 0 1 413.76 363.12 Tm <3120> Tj 1 0 0 1 421.44 365.52 Tm <2D20> Tj 1 0 0 1 431.52 362.88 Tm <6920> Tj 1 0 0 1 438.00 362.88 Tm <616E6420> Tj 1 0 0 1 458.40 365.28 Tm <2D3120> Tj 1 0 0 1 471.36 363.84 Tm <2B20> Tj 1 0 0 1 480.96 362.88 Tm <6920> Tj 1 0 0 1 487.44 362.64 Tm <617265> Tj 0.24 Tc 1 0 0 1 87.84 349.44 Tm <626F746820> Tj 1 0 0 1 112.80 349.68 Tm <636F6D706C657820> Tj 1 0 0 1 157.44 349.68 Tm <636F6E6A75676174657320> Tj 1 0 0 1 211.44 349.20 Tm <6F6620> Tj 1 0 0 1 224.40 349.44 Tm <74686520> Tj 1 0 0 1 242.64 349.20 Tm <73616D6520> Tj 1 0 0 1 269.52 349.20 Tm <6E756D6265722C20> Tj 1 0 0 1 312.24 348.96 Tm <7468657920> Tj 1 0 0 1 336.72 348.96 Tm <61726520> Tj 1 0 0 1 354.24 348.96 Tm <646566696E6974656C7920> Tj 1 0 0 1 403.20 348.96 Tm <6E6F7420> Tj 1 0 0 1 422.88 348.96 Tm <6571756976616C656E7420> Tj 1 0 0 1 475.92 348.72 Tm <746F20> Tj 1 0 0 1 488.40 348.72 Tm <65616368> Tj 0.24 Tc 1 0 0 1 88.08 335.76 Tm <6F746865722E20> Tj 1 0 0 1 119.28 335.76 Tm <54686520> Tj 1 0 0 1 141.36 335.76 Tm <616476616E7461676520> Tj 1 0 0 1 192.00 335.52 Tm <6F6620> Tj 1 0 0 1 205.44 335.52 Tm <636F6D706C657820> Tj 1 0 0 1 250.08 335.28 Tm <636F6E6A75676174657320> Tj 1 0 0 1 304.08 335.28 Tm <697320> Tj 1 0 0 1 315.12 335.04 Tm <7468617420> Tj 1 0 0 1 336.48 335.04 Tm <696620> Tj 1 0 0 1 347.04 335.04 Tm <6120> Tj 1 0 0 1 355.20 335.04 Tm <636F6D706C657820> Tj 1 0 0 1 399.36 335.04 Tm <6E756D62657220> Tj 1 0 0 1 440.64 335.04 Tm <697320> Tj 1 0 0 1 451.44 335.04 Tm <6D756C7469706C69656420> Tj 1 0 0 1 503.52 334.80 Tm <6279> Tj 0.24 Tc 1 0 0 1 88.08 321.84 Tm <69747320> Tj 1 0 0 1 102.72 321.84 Tm <636F6D706C657820> Tj 1 0 0 1 147.60 321.84 Tm <636F6E6A75676174652C20> Tj 1 0 0 1 199.44 321.60 Tm <74686520> Tj 1 0 0 1 216.96 321.84 Tm <70726F64756374697320> Tj 1 0 0 1 267.84 321.60 Tm <616C7761797320> Tj 1 0 0 1 304.32 321.36 Tm <6120> Tj 1 0 0 1 313.44 321.60 Tm <7265616C20> Tj 1 0 0 1 336.00 321.36 Tm <6E756D6265722E> Tj 0.24 Tc 1 0 0 1 87.84 294.00 Tm <546F20> Tj 1 0 0 1 104.40 294.00 Tm <64697669646520> Tj 1 0 0 1 138.00 294.00 Tm <74776F20> Tj 1 0 0 1 158.88 294.00 Tm <636F6D706C657820> Tj 1 0 0 1 202.80 293.76 Tm <6E756D626572732C20> Tj 1 0 0 1 250.32 293.76 Tm <776520> Tj 1 0 0 1 267.36 293.52 Tm <666972737420> Tj 1 0 0 1 289.20 293.52 Tm <777269746520> Tj 1 0 0 1 317.28 293.28 Tm <74686520> Tj 1 0 0 1 335.28 293.28 Tm <6469766973696F6E20> Tj 1 0 0 1 377.04 293.28 Tm <617320> Tj 1 0 0 1 390.24 293.28 Tm <6120> Tj 1 0 0 1 398.64 293.28 Tm <6672616374696F6E2C20> Tj 1 0 0 1 443.28 293.28 Tm <616E6420> Tj 1 0 0 1 463.44 293.28 Tm <7468656E> Tj 0.24 Tc 1 0 0 1 87.60 280.08 Tm <6D756C7469706C7920> Tj 1 0 0 1 132.24 280.08 Tm <74686520> Tj 1 0 0 1 149.52 280.08 Tm <6E756D657261746F7220> Tj 1 0 0 1 201.84 279.84 Tm <616E6420> Tj 1 0 0 1 222.24 279.84 Tm <64656E6F6D696E61746F7220> Tj 1 0 0 1 285.60 279.60 Tm <627920> Tj 1 0 0 1 300.48 279.60 Tm <74686520> Tj 1 0 0 1 318.72 279.60 Tm <636F6D706C657820> Tj 1 0 0 1 363.60 279.60 Tm <636F6E6A756761746520> Tj 1 0 0 1 413.76 279.36 Tm <6F6620> Tj 1 0 0 1 426.96 279.36 Tm <74686520> Tj 1 0 0 1 444.96 279.36 Tm <64656E6F6D696E61746F722E> Tj 0.24 Tc 1 0 0 1 87.84 266.40 Tm <576520> Tj 1 0 0 1 108.00 266.40 Tm <666F696C20> Tj 1 0 0 1 127.92 266.16 Tm <6F757420> Tj 1 0 0 1 146.16 266.40 Tm <74686520> Tj 1 0 0 1 163.44 266.16 Tm <746F7020> Tj 1 0 0 1 181.92 266.16 Tm <616E6420> Tj 1 0 0 1 201.84 265.92 Tm <626F74746F6D2C20> Tj 1 0 0 1 242.64 265.92 Tm <616E6420> Tj 1 0 0 1 262.80 265.92 Tm <736570617261746520> Tj 1 0 0 1 304.56 265.68 Tm <74686520> Tj 1 0 0 1 322.56 265.68 Tm <7265616C20> Tj 1 0 0 1 343.44 266.16 Tm <7061727420> Tj 1 0 0 1 365.76 265.68 Tm <66726F6D20> Tj 1 0 0 1 391.92 265.44 Tm <74686520> Tj 1 0 0 1 410.16 265.68 Tm <696D6167696E6172792C20> Tj 1 0 0 1 466.08 265.44 Tm <616E6420> Tj 1 0 0 1 486.00 265.44 Tm <65787072657373> Tj 0.24 Tc 1 0 0 1 87.84 252.24 Tm <6F757220> Tj 1 0 0 1 107.04 252.24 Tm <616E7377657220> Tj 1 0 0 1 144.48 252.24 Tm <696E20> Tj 1 0 0 1 156.48 252.24 Tm <74686520> Tj 1 0 0 1 174.00 252.24 Tm <666F726D20> Tj 1 0 0 1 200.40 252.24 Tm <6120> Tj 1 0 0 1 208.56 253.20 Tm <2B20> Tj 1 0 0 1 218.16 252.00 Tm <62692E> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 132.00 228.24 Tm <45782E20> Tj 1 0 0 1 151.68 228.72 Tm <283120> Tj 1 0 0 1 163.92 229.20 Tm <2B69292B20> Tj 1 0 0 1 193.44 228.48 Tm <287A2D69292D20> Tj 1 0 0 1 239.76 236.16 Tm <6C2B6920> Tj 1 0 0 1 266.64 232.56 Tm <2D20> Tj 1 0 0 1 280.32 236.16 Tm <6C2B6920> Tj 1 0 0 1 304.56 229.68 Tm <2A322B6920> Tj 1 0 0 1 340.80 232.32 Tm <2D20> Tj 1 0 0 1 353.28 235.92 Tm <6C2D337420> Tj 1 0 0 1 384.48 232.32 Tm <2D> Tj 1.50 Tc 1 0 0 1 238.80 217.68 Tm <322D6920> Tj 1 0 0 1 279.12 217.44 Tm <322D6920> Tj 1 0 0 1 312.96 217.44 Tm <322B7420> Tj 1 0 0 1 363.12 217.44 Tm <35> Tj 1.50 Tc /F3 13.0 Tf 1 0 0 1 398.16 235.92 Tm <4920> Tj 1 0 0 1 419.04 235.92 Tm <332E> Tj 1.50 Tc /F3 9.0 Tf 1 0 0 1 396.00 230.64 Tm <2D20> Tj 1 0 0 1 408.96 229.92 Tm <2D20> Tj 1 0 0 1 417.84 230.64 Tm <2D74> Tj 1.50 Tc /F3 13.0 Tf 1 0 0 1 397.20 217.20 Tm <3335> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 451.92 235.68 Tm <6C3369> Tj 1.44 Tc 1 0 0 1 435.12 227.52 Tm <6F7220> Tj 1 0 0 1 449.76 230.64 Tm <2D2D2D> Tj 1.50 Tc 1 0 0 1 450.72 217.20 Tm <3335> Tj 0.24 Tc /F3 11.0 Tf 1 0 0 1 87.36 190.08 Tm <4D756C7469706C69636174696F6E20> Tj 1 0 0 1 159.84 190.08 Tm <616E6420> Tj 1 0 0 1 179.76 190.08 Tm <6469766973696F6E2C20> Tj 1 0 0 1 224.88 189.84 Tm <617320> Tj 1 0 0 1 237.60 189.84 Tm <77656C6C20> Tj 1 0 0 1 261.36 189.84 Tm <617320> Tj 1 0 0 1 274.32 189.84 Tm <736F6D6520> Tj 1 0 0 1 302.64 189.60 Tm 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<436F6D706C657820> Tj 1 0 0 1 282.00 148.08 Tm <4E756D62657273> Tj 0.24 Tc 1 0 0 1 87.12 120.96 Tm <416E7920> Tj 1 0 0 1 111.36 120.96 Tm <636F6D706C657820> Tj 1 0 0 1 155.76 120.96 Tm <6E756D6265722C20> Tj 1 0 0 1 198.00 120.72 Tm <7A3A20> Tj 1 0 0 1 216.48 120.72 Tm <6120> Tj 1 0 0 1 224.64 121.44 Tm <2B20> Tj 1 0 0 1 234.24 120.48 Tm <62692C20> Tj 1 0 0 1 249.84 120.48 Tm <63616E20> Tj 1 0 0 1 269.28 120.48 Tm <626520> Tj 1 0 0 1 283.44 120.48 Tm <7772697474656E20> Tj 1 0 0 1 321.12 120.24 Tm <696E20> Tj 1 0 0 1 333.60 120.24 Tm <69747320> Tj 1 0 0 1 348.24 120.24 Tm <6571756976616C656E7420> Tj 1 0 0 1 401.04 120.24 Tm <747269676F6E6F6D657472696320> Tj 1 0 0 1 471.84 120.00 Tm <666F726D2C20> Tj 1 0 0 1 499.68 119.76 Tm <7A20> Tj 1 0 0 1 508.08 122.40 Tm <3A> Tj 0.24 Tc 1 0 0 1 86.88 106.80 Tm <722A28636F7320> Tj 1 0 0 1 120.72 106.80 Tm <6420> Tj 1 0 0 1 129.60 107.76 Tm <2B20> Tj 1 0 0 1 139.44 106.80 Tm <692A73696E20> Tj 1 0 0 1 165.36 106.80 Tm <39292E20> Tj 1 0 0 1 180.96 106.56 Tm <496E20> Tj 1 0 0 1 193.68 106.56 Tm <7468697320> Tj 1 0 0 1 214.80 106.56 Tm <666F726D2C20> Tj 1 0 0 1 243.84 106.32 Tm <7220> Tj 1 0 0 1 251.04 106.32 Tm <697320> Tj 1 0 0 1 261.84 106.32 Tm <74686520> Tj 1 0 0 1 279.36 106.32 Tm <6D61676E69747564652C20> Tj 1 0 0 1 337.92 106.08 Tm <616E6420> Tj 1 0 0 1 357.12 106.08 Tm <3020> Tj 1 0 0 1 365.76 106.08 Tm <697320> Tj 1 0 0 1 376.56 105.84 Tm <74686520> Tj 1 0 0 1 394.56 105.84 Tm <617267756D656E742E> Tj ET endstream endobj 4 0 obj 18530 endobj 5 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 6 0 obj << /Length 7 0 R >> stream Complex NumbersIntroduction to Complex NumbersThe idea of complex (or imaginary) numbers is very useful in many areas. Our discussionof these numbers begins with defining i : J- I ro that i2: -1. In general (but notalways), complex n-umbers have a real part and an imaginary part. They are written inseveral forms. the most common beinez: a*biwhere z is the whole complex number, a is the real part, and bi is the imaginary part.Operations on Complex NumbersAddition & Subtraction: Two complex numbers, a + bi and c + di, are added andsubtracted by simply combining the real parts and imaginary parts separately, as follows:(a+ bi) + (c + di) : (a + c) + (b + d)i(a + bi) - (c + di) : (a - c) + O - d)iMultiplication: Two complex numbers are multiplied using a process just like F.O.LL.(a+ bi) * (c + di) : ac + adi + bci + bdi2 : (ac -bd) + (ad + bc)iDivision: To divide complex numbers, we need to first learn complex conjugales. Twocomplex numbers are'called complex conjugates if they have the same real part and thesztme imaginarypart, with only one of the signs changed. (It is far more common tochange the sign of the imaginary part, but either will work). Thus, I + i and I - i arecomplex conjugates, and so are 1 + i and -1 + i. Note that although 1 - i and -1 + i areboth complex conjugates of the same number, they are definitely not equivalent to eachother. The advantage of complex conjugates is that if a complex number is multiplied byits complex conjugate, the productis always a real number.To divide two complex numbers, we first write the division as a fraction, and thenmultiply the numerator and denominator by the complex conjugate of the denominator.We foil out the top and bottom, and separate the real part from the imaginary, and expressour answer in the form a + bi. Ex. (1 +i)+ (z-i)- l+i - l+i *2+i - l-3t -2-i 2-i 2+t 5 I 3.- - -t35 l3ior ---35 Multiplication and division, as well as some other operations are much more easilyperformed using other forms of complex numbers, as we will see later on.The Trigonometric Form of Complex NumbersAny complex number, z: a + bi, can be written in its equivalent trigonometric form, z :r*(cos d + i*sin 9). In this form, r is the magnitude, and 0 is the argument. endstream endobj 7 0 obj 2235 endobj 8 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 9 0 obj << /Length 44 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj2 Do Q endstream endobj 10 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj2 2 0 R >> /Subtype 6 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 8 0 R 9 0 R 3 0 R ] >> endobj 13 0 obj << /Type /XObject /Subtype /Image /Width 2551 /Height 3299 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /Columns 2551 >> /Length 42888 >> stream ]W\w_m7U+}msR}]km+^ 7ka&W{T_l4]0II4ӆMx>'iCA 7kM6(4JAմavai@D]3N a0]Pa5&>"#""PB" DGSɸ菓X*|Y~Nhh-ܦOӏU31Fu0 h2r3gAtKĆ? qAL!g ̐`Aa 0Ij 5p#/;z.P7q2۴l l պt '}-Ϻ;PׯuSm{B[^z?c/1]{9"]`Q^Lw}aF?۷]o'{u/VHGa'AZM]Zamm/j[^E7W~# |ڻV[m$ᠯ5b0NL ݄&q 4 ph tAh0Xj DDDDDDDDDDYξeVy :.'kb:ARl;T:W:Y٪ίz.4uܕ-h7NKjWxU؊!MQ֖KzAp\.-B (Aa}UI|CIr Ц47Oױ +4:OO"/|-Kizz]ZlzX[/DG McƑ(qܨ uu-%^V?a_ízuOzAY}xj%iA7z.QƷKb=.-++ ◥a4LݧL Rk{iC eL ʃ [ EEDDDD\DFL ݊i'Dv> 0E9""#JV҆҈gde?x)Pa9 ɎA IXa>憋O=*~Z;~Z%;--$Z`>i-A>H29BH4Хc_^W_ZE} AgUKA%(`b /Rb 3I z Dhv2LΨI O6P0f` U,;PPL'=*tk($=P]K4҆@K^,dvƝW\pB]$BA}U7MKT"_z^ֿY"0R8sk֗Џa_n^KKլ7_VM,7uLzUY3ï}Uk'wÊZAUz:޲1a4i8a+ӊ"X@Aڴ4kϸ h5iDDDF5ZUȨO ?l KD3.*H*K_IqI!%ֽ-]-xuTT{ڤZK 2ܬd&YنrZ8`BB  oVUv@Lv iAe)qHw< *kA`.+H|!h1a 0`WA0L8pt^\LO¨T1=[ȢiJ U_\t\Qq ~|4+uVXzf-Wӿ~?Wk߯/rbʗU%kBs;Ό&]f CO_® Z 0@3  Es!;6A28fĽ*Ax rC$M0w gAA݂33W3PQ>2@ˢA:d#vUlXAkN$p L&ӏPA5DO0Aȸ,!a''f$CHfǜ |`0DBR.SY{@վނ m6 2T^MHQ/$8- DŢwN4P&*B{Ҡ.0L&jf(L ,ᙙ!GoW i=`%PN`ȹq A ɞ NФE 0Anɲ{iZ??OJ  H}TzM/дkk[^xH9CP~~XT7,j Ƀ__X^u׏_WZZ1_/ߊ/U!@dI&FK뮕ܴr࿰nO_a@Zun*? n}$ԨJ}nK"Qu:"tf0m]!Ed5RWa@_ڻa0c_յ_!$i~ _V;U//C|a0[ w8ֺo?|0{ƒ ; f x`ҽҾOv ? ~m65Tb;=5mNV|5^m[tnbBMbӯV\E Պ`+a;#,+~SNv DAM  ;b4RIVHH &GQ_{Om,mDph4AL&Ml(+iI 0pMNUE|C`I;aS1Ša2X a,a|A,B,=5ݚ;m;M":؃0pA A060خLHxDDDDDDDDDDDDDA4)7Rӆ00L(L&dX$?Z /Z ֩MV2#ڈhD20L=~T+ ,Lw Y@07L' pDDDDDDDDDR;Z0_DFK[xaB R$0 fA4a&IOxMӠUIkZK]RTJT"$KhU K\dc5 (LJYdRDA42a)fIVÇH&i(\Ղy FKptq S(*at_4 ӤW I&vPq}E(A &B $&Iݵ ®~ xI}dLCΑ3"c<<2:>3HqH QQ kpA%9Q5L L~k  j0L* όx aD;#//kƥ>H?/59 :P0B„/ C N!0ia a|a4 n@Iy gAN#䁓A2{#FEB:#\@,% Ao8ENH6.蝴-Ua N4a Մ h X]B 4C4 vC"A t4@"CӤd|vBCv `"EEuNzpd\v LiD<"Q h;O A4xLv]4z&NFf`[8%L,0J*itWK^K_`H 69m%$nN ƉD4'x*DiwDa0PTuuoUmkݧkmd# ? 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0 R >> stream Complex Numbers The conversions are: 0:tarL-t (bla),: J;\b' x:r*cosdy:r*sind *Note: Be careful when finding d, because tan-l is only define d for -!.0.L ,but22complex numbers can be anywhere from 0< 0< 2 n. Because of this, we should think ofthe answer from the calculator as the reference angle, and use what we know about thesigns of the real and imaginary parts.As an example, z: -l+ i is in the second quadrant, Uut tan-I1{): -45o, which is in the'-l'fourth quadrant and cannot be our angle. If we use the reference angle, 45o, and the factthat the angle we want is in the second quadrant, we get 180o-45o : 135o.The Exponential FormThere is a third form used to represent complex numbers called the exponential form, andit is related to the trigonometric form by Euler's Identity: et' : "os g + i*sin d. so reio:r(cosd+i*sind) This form is especially useful when we want to raise complex numbers to largeexponents. For example, (1 + i)100 would be extremely cumbersome, even with the binomial theorem. If we instead convert (1 + D to its exponential form, r:42 and 0 : tar[' (l\ : tr 14 (l + i)100 : (J, e|/t t4)roo - 2so t'etzslt - zso *"i17 -2soa(-l) : -250, which is much easier to work with.It is very difficult to try to add or subtract complex numbers in this form, butmultiplication and division are very easy. In multiplication, multiply the magnitudes andadd the arguments. In division, divide the magnitudes and subtract the arguments.There is another useful formula involving the exponential form known asDeMoivre's Theorem. It says(cos d + i*sin 0)n: cos (nd) + i sin (nB)This relation is due to the fact that (cos 0 * i*sin d)n :1eid;" -.indThis can be used to find equivalent expressions for functions such as cos (3d).cos 0 is the real part of cos 0 + i*sin 0, so cos 39 is the real partof (cos 0^+ i*sin d)3The right side can be expanded using FOIL, and we will get cos 3d = cos' d -3 cos d sin0. endstream endobj 17 0 obj 1918 endobj 18 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 19 0 obj << /Length 45 >> stream q 612.24 0 0 791.76 0.00 0.00 cm /Obj13 Do Q endstream endobj 20 0 obj << /Type /Page /MediaBox [ 0 0 612.24 791.76 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj13 13 0 R >> /Subtype 16 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 18 0 R 19 0 R 14 0 R ] >> endobj 11 0 obj << /Kids [ 10 0 R 20 0 R ] /Count 2 /Type /Pages >> endobj 12 0 obj << /Pages 11 0 R /Type /Catalog >> endobj xref 0 21 0000000000 65535 f 0000000015 00000 n 0000000110 00000 n 0000051054 00000 n 0000069638 00000 n 0000069660 00000 n 0000069763 00000 n 0000072051 00000 n 0000072072 00000 n 0000072150 00000 n 0000072245 00000 n 0000133666 00000 n 0000133739 00000 n 0000072494 00000 n 0000115601 00000 n 0000131217 00000 n 0000131240 00000 n 0000133213 00000 n 0000133235 00000 n 0000133314 00000 n 0000133411 00000 n trailer << /Size 21 /Info 1 0 R /Root 12 0 R /ID [] >> startxref 133794 %%EOF