In a sense, all power series of finite positive radius of convergence are a variant of the geometric series. Then f(z)±g(z) and f(z)·g(z) also have convergent power series expansions, which are (at least) convergent in the same disk |z|< r.
2 Simple Fourier series in the Theory of Complex Functions
If we use polar coordinates here, z=r eiθ, then we get for each xedr∈]0, [ a Fourier series of the function φ(θ) given by. When f(z) = 1/(1−z),|z| < 1, is treated in the same way, we obtain after some calculations the following important formula.
3 Power series
It follows that the radius of convergence is 1. withn2 in the exponent, it would not be a good idea to use the criterion of quotients. Example 3.6 Find the radius of convergence for each of the series. a) Sincecn=nn, we get by the measure of roots that 1.
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Ifa∈]0,1], then it follows from the first equality sign that the radius of convergence r= 1. Calculate the radius of convergence for the series +∞. It follows that the radius of convergence is w = lim.
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4 Analytic functions described as power series
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5 Linear differential equations and the power series method
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6 The classical differential equations
If the power series solution is not a polynomial, (14) means that the radius of convergence is 1. Since (n+ 2)(n+ 1)= 0 in the summation domain N0, we get the following recursion formula by the identity theorem ,.
7 Some more difficult differential equations
Becausen+ 1= 0 in the summation domainN0 we get the following recursion formula from the identity theorem. Reach your full potential at the Stockholm School of Economics, in one of the most innovative cities in the world.
8 Zeros of analytic functions
Here it is difficult – although not completely impossible – to insert the power series expansions, so we prefer the differentiation method here. Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more Click on the ad to read more.
9 Fourier series
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10 The maximum principle
