3. Find all entire functions f(z) satisfying the condition (Hint: What can you say about f' (z)?) 3. Find all entire functions f(z) satisfying the condition (Hint: What can you say about f&#...
5. Find all entire functions f(z) such that f(2 z) on C. JuSti your answer. (Hint: C is uncountable.)
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
Suppose fis an entire function such that there is a number M such that Re(f(z)2 - Imf(z))2 s M for all z. Prove that f must bie Hint: Compare to exercises #8: if f and g are both entire, then so are f-g and gof. Find an appropriate g so that you may apply Liouville's theorem to the entire function gof
5. (a) Find all complex z satisfying 24 + 16 = 0. (b) Find the inverse Laplace transform of F(s) = 116 using the inversion formula 8(e) = 221 /** *F(2)dz 2ni Jo-100 and the Cauchy residue theorem. Indicate for which values of o the above is valid. Describe clearly the contour you are using.
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
2. For each of the following functions X(), what can you say about x(t)? In other words, what are x() and ( t o)? Is x(t) converging or diverging? Is x(t) oscillatory? 6(3+2) X(s)=– (s? +95 +20)(s+4) 6(3+2) X(S)= (s? +45 +3)(3+4) W o 20= +4*3% +) () – 165.75 X(S) = 16s+5 32 +9
5. Let f(z) = arctan(z) (a) (3 marks) Find the Taylor series about r)Hint: darctan( You may assume that the Taylor series for f(x) converges to f(x) for values of r in the interval of convergence (b) (3 marks) What is the radius of convergence of the Taylor series for f(z)? Show that the Taylor series converges at z = 1 (c) (3 marks) Hence, write as a series. (d) (3 marks) Go to https://teaching.smp.uq.edu.au/scims Calculus/Series.html. Use the interactive animation...
In(z) 3, Consider the function f(x)= (a) Find the Taylor series for r(z) at -e. b) What is the interval of convergence for this Taylor series? (c) Write out the constant term of your Taylor series from part (a). (Your answer should be a series!). (d) What can you say about the series you found in part (c), by interpreting it as the limit of your series as x → 0. (Does it converge? If so, what is the limit?)...
x(f(y) = x))) (Hint: the rules about functions in first order logic apply to functions in set theory also) 3. Prove that for any set X, the function f : X → P(X) where f(x)-{y E X : yメx} is an injection. x(f(y) = x))) (Hint: the rules about functions in first order logic apply to functions in set theory also) 3. Prove that for any set X, the function f : X → P(X) where f(x)-{y E X :...
Let V be the vector space consisting of all functions f: R + R satisfying f(x) = a exp(x) +b cos(x) + csin(x) for some real numbers a, b, and c. (The function exp refers to the exponential, exp(22) = e.) Let F be the basis (exp cos sin of V. Let T :V + V be the linear transformation T(f) = f + f' + 2f" (where f' is the derivative of f). You may use the linearity of...