LECTURE NOTES FOR MATHEMATICAL ANALYSIS FOR PHYSICS

1. KWAME NKRUMAH UNIVERSITY OF. SCIENCE AND TECHNOLOGY.

2. Fourier Analysis for Periodic Functions. The Fourier series representation of analytic functions is derived from Laurent expansions. The elementary complex analysis is used to derive additional fundamental results in the harmonic analysis including the representation of C∞ periodic functions by Fourier series, the representation of rapidly decreasing functions by Fourier integrals, and Shannon’s sampling theorem. The ideas are classical and of transcendent beauty..

3. consisting of 2L-periodic functions converges for all x, then the function to which it converges will be periodic of period 2L. There are two symmetry properties of functions that will be useful in the study of Fourier series..

4. Examples:. Sums of odd powers of x are odd: 5x3 − 3x.

5. Typically, f(x) will be piecewise-defined.. Big advantage that Fourier series have over Taylor series: the function f(x) can have discontinuities..

6. For example the figure below shows the periodic extensions for.

7. Applications. A Fourier Series has many applications in mathematical analysis as it is defined as the sum of multiple sines and cosines. Thus, it can be easily differentiated and integrated, which usually analyses the functions such as saw waves which are periodic signals in experimentation. It also provides an analytical approach to solve the discontinuity problem. In calculus, this helps in solving complex differential equations..

8. The above Taylor series expansion is given for a real values function f(x) where f’(a), f’’(a), f’’’(a), etc., denotes the derivative of the function at point a. If the value of point ‘a’ is zero, then the Taylor series is also called the Maclaurin series..

9. Proof We know that the power series can be defined as f(æ) anxn When x = O, f(x)= ao So, differentiate the given function, it becomes, f'(x) = al+ 2a2X + 3a3X2 + 4a4X3 Again, when you substitute x = O, we get f'(O) So, differentiate it again, we get f"(x) = 2a2 + 6a3X +12a4X2 + Now, substitute x=0 in second-order differentiation, we get = 2a2 Therefore, = a2 By generalising the equation, we get f n (0) / n! = an Now substitute the values in the power series we get, fm(0) 3 f(æ) = f(O) + f + 2! Generalise f in more general form, it becomes f(x) = b + bl (x-a) + x-a)2 + b3 (x-a)3+ . Now, x = a, we get bn = ft(a) / n!.

10. Applications of Taylor Series. The uses of the Taylor series are:.

11. • Taylor series is used to evaluate the value of a whole function in each point if the functional values and derivatives are identified at a single point..

12. f’’(x) = -cos x ⇒ f’’(0) = -1. f’’’(x) = sin x ⇒ f’’’(0) = 0.

13. Power Series. A power series is a special type of infinite series representing a mathematical function in the form of an infinite series that either converges or diverges. Whenever there is a discussion of power series, the central fact we are concerned with is the convergence of a power series. The convergence of a power series depends upon the variable of the power series..

14. o Nowhere convergent – if the power series is not convergent for any value of x.

15. Solved Examples on Power Series. Example 1:. Find the radius of convergence of the series x + x2/22 + (2!/33)x3 + (3!/44)x4 + …...

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17. SUMMARY ON POWER SERIES. What is a Power series?.

18. The radius of convergence of a power series is a real number 0 ≤ R ≤ ∞, such that if |x| R then the power series diverges..