Cardinality and Invariant Subspaces by L. de Branges

By L. de Branges

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5 Prove that 6 Plot the points 0, 1, e^, 1 + e^ on an Argand diagram. What sort of quadrilateral do these points form? 6 and Question 5. 7 Prove that 8 If z = f(p) + jg(p)> where p is a real parameter, then the derivative and integral of z with respect to p are defined by and Prove that if z = e°^ where a is a fixed complex number, then and 9 The position at time t of a point Z moving in an Argand diagram is given by z = re j , where r and 9 depend on t. Find and and deduce the radial and transverse components of the velocity and acceleration of Z.

EXERCISE 2B In this exercise you should make full but critical use of a graphic calculator or computer if these are available. 1 Make a table of values of 8 sin 9 for 9 from 0 to n at intervals of and say what happens when n < 9 < 2n. By plotting points draw the curve r = 8 sin 9. Prove that this curve is a circle, and give its cartesian equation. 2 Draw the graph of the spiral of Archimedes 26 3 A curve with polar equation r = k sin n9> where k and n are positive and n is an integer, is called a rhodonea (rose curve).

Proof The proof is in three parts, in which n is (i) positive (ii) zero or (iii) negative. (i) When n is a positive integer the proof is by induction. The theorem is obviously true when n = 1, and if (cos 0 + j sin 9)k = cos k9 + j sin k9 then (cos 0 + j sin 0) fc+1 = (cosfc0+ j sin k9)(cos 9 + j sin 0) = cos(fc0 + 0 ) + j s i n ( f c 0 + 0) = cos(fc+l)0 + jsin(fc+l)0 so by induction the theorem is true for all positive integers n. (ii) By definition, z° = 1 for all complex numbers z ^ 0. Therefore (cos 0 + j sin )° = 1 = cos 0 + j sin 0.

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