Review of Mathematical Concepts

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Elmer Allison
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1 ENEE 322: Signls nd Systems view of Mthemticl Concepts This hndout contins ief eview of mthemticl concepts which e vitlly impotnt to ENEE 322: Signls nd Systems. Since this mteil is coveed in vious couses which e peequisites fo ENEE 322, no clss time will e devoted to this infomtion. If you hve wek ckgound o do not fully undestnd this hndout, plese consult the efeences nd see the TA o pofesso duing office hous. Complex Numes A complex nume is n expession of the fom z + j () whee nd e el numes nd j is defined s j. We sy tht is the el pt of the complex nume, is the imginy pt of the complex nume, {z} {z} nd () is the ectngul fom of the complex nume. A complex nume cn e viewed s point in the xy-pne which is then efeed to s the complex plne (see Fig. ). Any point in the xy-plne cn e descied y eithe ectngul o pol coodintes (see Fig. ), so let (, ) e the pol coodintes fo z + j nd note tht Using these convesions in () we see tht cos sin ctn ( ) (2). z + j cos + j sin (cos + j sin ). (3) To simplify this expession futhe, ecll Eule s identity e j cos + j sin (4) which cn e veified y comping the seies expnsion of e j, cos, nd sin. Using (4) in (3) yields the mgnitude nd phse fom of complex nume, z e j z e j z (5) whee we hve used z to denote the mgnitude nd z to denote the phse of z. Eule s identity (4) cn lso e used to show the following eltionships: e j cos + j sin cos ej + e j 2 e j cos j sin sin ej e j j2 (6)

2 ENEE 322: Signls nd Systems 2 The conjugte of complex nume z is denoted y z nd defined s (see Fig. 2) z + j z j z e j z e j. (7) A few useful eltionships involving the conjugte include z + z 2 z z j2 ( + j) + ( j) 2 ( + j) ( j) j2 {z} {z} zz ( + j)( j) 2 j + j z 2 z z z z z z ej e j e j2 e j2 z z + w 2 z w 2 4 {zw } z 2 w 2 {(z + w)(z w) } The ddition of two complex numes is pefomed y dding thei espective components z + z 2 ( + j ) + ( 2 + j 2 ) ( + 2 ) + j ( + 2 ). (8) To multiply two complex numes which e in ectngul fo, ecll tht j 2 nd multiply s usul z z 2 ( + j ) ( 2 + j 2 ) (9) 2 + j 2 + j 2 + j 2 2 Howeve, multipliction tends to e esie in pol fom 2 + j 2 + j 2 2 z z 2 ( 2 2 ) + j ( ). (0) z z 2 ( e j ) ( 2 e j 2) ( 2 ) e j( + 2 ). () So when two complex numes e multiplied, thei mgnitudes e multiplied nd the phses e dded. Complex ddition nd multipliction e shown in Figs. 3 nd 4, espectively. Finlly, the tingle inequlity holds fo ny two complex numes z nd z 2 z + z 2 z + z 2 (2) z 2 z z 2 z (3) A ief eview of complex numes cn e found in [, 2]. An intoductoy level tetment of the mteil cn e found in [3, 4], nd igoous tetment is in [5, 6, 7].

3 ENEE 322: Signls nd Systems 3 2 Summtions Thoughout this couse we will e evluting summtions of the fom n N fo some possily complex nume. To find simple closed-fom solution, fist conside the cse whee. Since n, we hve N. (4) Fo, wite out the tems of the sum nd conside multiplying y N N + N If we sutct these two expessions, the intemedite tems cncel ech othe out. ( n ) ( ) n N () N N (5) Putting (4) nd (5) togethe we hve N, N, (6) This fundmentl esult cn e used to deive mny othe summtions. Fo exmple, conside modifying the summtion to hve ity limits N nd N 2. To evlute the summtion, pefom the chnge of viles m n N to get it into the fom of (6), N 2 nn N 2 N m0 m+n N N 2 m0 m N 2 N +, N N 2 N +,

4 ENEE 322: Signls nd Systems 4 Hence N 2 nn N 2 N + N N 2+, Altentively, (7) could hve een deived y splitting the summtion into two pts. Fo, ( ) ( ) N 2 N 2 n N n n N 2 + N N N 2+ nn Now if we estict to hve <, the uppe limit of (6) cn go to infinity, ( ) N n lim n N lim. N N Becuse n 0 s N when <, this educes to Now view (8) s function of nd diffeentite oth sides Hence, ( d ) n d ( ) d d n n n n n d d ( () 2 () 2 () 2 (7) < (8) ) () 2 < (9) This poceedue cn e epeted to solve summtions of the fom n k n. Finlly, ecll tht switching the ode of the limits of n integl chnges the esult Howeve, fo summtions nd f(t)dt f(t)dt. N 2 nn N + N N 2 N N + + N Moe summtions cn e found in [8] N nn 2 n.

5 ENEE 322: Signls nd Systems 5 feences [] B. P. Lthi, Signl Pocessing nd Line Systems. Cmichel, Clifoni: Bekeley-Cmidge Pess, 998. [2] J. D. Sheick, Concepts in Systems nd Signls. New Jesey: Pentice Hll, 200. [3] E. B. Sff nd A. D. Snide, Fundmentls of Complex Anlysis fo Mthemtics, Science, nd Engineeing. New Jesey: Pentice Hll, 993. [4] R. V. Chuchill nd J. W. Bown, Complex Viles nd Applictions, 5-th ed. New Yok: McGw Hill, 990. [5] L. V. Ahlfos, Complex Anlysis. New Yok: McGw Hill, 953. [6] E. Hille, Anlytic Function Theoy, Vol. I, 2nd ed. New Yok: Chelse, 973. [7] J. W. Dettmn, Applied Complex Viles. New Yok: Dove Pulictions (pint), 984. [8] CRC Stndd Mthemticl Tles nd Fomule (30th Ed). CRC Pess, 996.

6 ENEE 322: Signls nd Systems 6 z + j sin() z + j exp(j) () ctngul coodintes cos( ) () Pol coodintes Figue : Complex Plne z + j exp(j) - z* - j exp(-j) Figue 2: Complex Conjugte z +d z + w d s φ c w +c Figue 3: Complex Addition z + j exp(j) d+c zw d s φ c w c + j d s exp(jφ) s + φ c-d Figue 4: Complex Multipliction

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Fourier-Bessel Expansions with Arbitrary Radial Boundaries Applied Mthemtics,,, - doi:./m.. Pulished Online My (http://www.scirp.og/jounl/m) Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk