Complex Practice Exam 1

Transcription

1 Complex Practice Exam This practice exam contains sample questions. The actual exam will have fewer questions, and may contain questions not listed here.. Be prepared to explain the following concepts, definitions, or theorems: A complex number, polar coordinates, rectangular coordinates Add, Multiply, Sub, Div, Conjugate, abs Value - graphical interpretations of these Complex roots, graphically and algebraically The limit of a complex function f() as approaches c is L Continuity of a complex function f() at a point = c The complex derivative of a function f() Analytic function and Entire function CR equations f() analytic & f () = 0, f() analytic & f-conjugate analytic, f() analytic and f() constant Harmonic function and harmonic conjugate of a function u (incl. how to find) Arg(), arg(), e, sin(), cos(), log(), and Log() Euler s Formula, De Moivre s Formula Complex parametric functions (t), their integrals and derivatives Different paths (line segments and circles) Contour Integrals. Describe the set of points such that (a) Re( ) (b) (c) Arg ( ) 4. Let i. Draw, in one coordinate system,,, 4, and

2 4. Compute/simplify the following and find real and imag parts: a) ( i )( i) i (b) i i ( i) (c) ( i) 6 (d) i i 5. Find the fourth roots of -, i.e. 4, and display them graphically. Do the same for the fifth roots of - and of (+i). 6. Consider the following questions, involving limits and continuity of complex functions. Remember that limits can be taken in different directions, and for complicated limits there is l'hospital's rule as long as the function is C-differentiable. a) If f ( x iy ) x, then f is clearly undefined at = 0. Can you define f(0) in such a iy way that the new function is continuous at every point in the complex plane? Hint: focus on the point (0,0).

3 9 i b) Say f ( ) Can you define f(i) in such a way that the new function 5 i is continuous at every point in the complex plane? 6 c) Find lim 0, lim 6 d) i 0 lim e) i Consider the following questions about analytic functions. a) If f ( ) then determine where, if at all, the function is analytic. If it is analytic, find the complex derivative of f. b) If f ( ) x xy i x y y then determine where, if at all, the function is analytic. If it is analytic, find the complex derivative of f.

4 8. Decide which of the following functions are analytic, and in which domain they are analytic. If a function is analytic, find its complex derivative: e (a) f ( ) e (b) f ( ) x ix y xy x iy iy x 9. Consider the function u( x, e sin(. Is it harmonic? If so, find its harmonic conjugate. Do the same for (a) u ( x, x xy xy y And for u( x, e cos( x) 0. Please find the following numerical answers: (a) e i

5 (b) cos( i) (c) sin i (d) log( ) (e) Log( i). Solve the following equations for. 4 (a) 0 (b) e (c) sin( ) i 4 (d) e (e) cos( ) isin( ). Use the definition of derivative to show that the functions f ( ) Re( ) is nowhere differentiable.

6 Use the CR equations to show that the function f ( ) is nowhere differentiable. Show that if v is the harmonic conjugate of u, then the product u v is harmonic. Prove that if h( x, is a harmonic everywhere then the complex function f ( ) h( x, i h( x, is an analytic function for all. x y 5 Show that e if Re( ) 0 6 State De Moivre s formula. Then use it to prove the trig identity sin( x) sin( x)cos( x) 7 Show that the function i e is periodic with period

7 8 Show that the function sin() is unbounded 9 Show that the function f ( ) x cannot be an analytic function. 0 Prove that sin ( ) cos ( ) (Hint: there are several ways to do this. One slick way involves taking the derivative of f ( ) sin ( ) cos ( ). Another possibility is to work with the actual definitions of sin and cos) Prove the following theorem: If f() is an analytic function with values that are always imaginary, then the function must be constant. Find complex parametric functions representing the following paths: (a) a straight line from i to i (b) the right half of a circle from i to i, (c) a straight line from - i to + i

8 (d) a circle centered at +i of radius i Evaluate a. '(t) for ( t) cos(t) isin(t ) 0 b. ( t) dt for ( t) (5 4i) e it 4 Evaluate a. i d where is a line segment from --i to +i b. d where is a circle radius centered at the origin c. d where is a circle radius centered at the origin 5 Is raised to any power (integer or otherwise) always equal to?