Physics 325 Lecture 1 - Physics 325 Lecture 1 (with thanks...

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Physics 325 Lecture 1 (with thanks to Jim Wiss and Aida El-Khadra) Series approximations Despite what you may have learned in Physics 21X, there are very few exactly solvable, real world problems in physics. Approximations therefore play a very important part. In particular, series approximations. The alternative in many cases, are numerical solutions using computers (which also have an important place). Taylor’s theorem: Take a well-behaved function, f(x), with derivatives that are finite. Taylor’s theorem says () ( ) ( ) () 0 00 2 2 0 0 2 11 2! n n n xx x df d f d f fx fx dx dx n dx = x = = =+ + + + where x 0 is an arbitrary point about which to expand. As an example, let’s expand f(x)=e x about x 0 =0 . 0 2 0 1 n n xn dd ee dx dx x ex x x n == =⇒ =+ + + + + …… This is quite useful because we see that when x<< 1, we can write 1 x + ± Another example: f(x)=sinx () () 2 2 0 23 3 sin cos sin sin . 0 sin 0 cos 1 1 sin 0 1 0 1 ! 3 ! x x e t c dx dx x xxx x x x = =+ − + =− + ii i Again, for small x sin x ~x. You can show for yourself that 24 cos 1 ! x = −+
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and cos x~ 1 for small x . The expansions can be useful as tools for integration as well. An example: Integrate 0 1 1 t dx x 2 23 0 1 1 1 11 1 12 3 t xx x dx t t t x =+ + + = +++ And furthermore, since () 1 ln 1 1 d x dx x −= we can write ln 1 x x ⎛⎞ −+ + + ⎜⎟ ⎝⎠ One more: f(x)=(1+x) n 2 2 0 2 ; 1 1 1 0 1 1 2 nn n n dd xnx xn n x dx dx x x n n x n +=+ += = + + + so, for small x n x nx + + ± Since 1 2!
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Physics 325 Lecture 1 - Physics 325 Lecture 1 (with thanks...

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