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hw1_solutions - Phgsics 325 Fall 2006 Solutions to Homework...

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Phgsics 325, Fall 2006 Solutions to Homework Assignment #I i) Taylor expansion of erf (r) : # f; e-"u'du. we need the first three nonzero terms, evaiuatedatr:0. erf(0) : g erf' - rtu:" by the fundamental theorem of calculus. erf, : h u, r : 0. erf't = rtl-Zr"-"] which vanishes at c:0, er.ft" : hl-r"-n + 4rr"-,"1which has-the value ff at r : 0. erfrrrt - ftl+re-,' *8xe-,2 - Brt"-,'l which ,runi.n", at r = 0. :: {""' ;,k [-tle-" + 4e-" + Be-,''- I6r2e-,' - 24rze-,' * r6rae-,rl which has r]re value # at r:0. The firit three nonzero terms of the Taylor expansion are e r f ( r ) = o * h * o - # + o + m Hete's an elegant alternative solution (easier, tool). you can express e-u, : Ino g#f , and integrate that ,:l.,T,l{,t.rm to get a series expansion for eri(x). erf(x):rtD|y)ffi Reading off the first'three terms gives e r f ( r ) = * 1 " - 4 + t l l Which agrbes' with it . u"U'ou. computation. Inserting r: I to the seriesexpansion, erf(I) = 0.g6b1, which can be compared with the value looked up in a table (looking up erf(l) in a table is not required for credit), erf (I) :0.8427.
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