Let Log(2) be the principal branch of the logarithm. Then lim (Log(iy – 1) – Log(-iy...
5. Let f(x) = e-} Log(z) (that is, f is the principal branch of z-1/2). Compute [flade, where (a) (2 points) y is the upper half of the unit circle C(0) from +1 to -1; (b) (2 points) y is the lower half of the unit circle C1(0) from +1 to -1.
2. Let f(z) be the principal branch of i.е., f(z) exp@ Log(z)}. Co mpute (e)dz where C is the semicircle {et : 0 < θ < π
Use log, 2=0.371, log, 3=0.55, and log, 50.804 to approximate the value of the given logarithm to 3 decimal places. Assume that b>0 and b#1. X 5 10%)
4. Let γ be a path with parametrization γ(1)-(1 12,1) where 2 1 £1. Compute lyidz. (Here 1/2 is understood by using principal branch of logarithm.) 4. Let γ be a path with parametrization γ(1)-(1 12,1) where 2 1 £1. Compute lyidz. (Here 1/2 is understood by using principal branch of logarithm.)
(6) Pretend you know that the natural logarithm function log : (0,00) + R exists and has the properties you remember, in particular log(24) = x log(2) for all x. Let p > 0. For what values of p does the series » - converge? log(n)P:n Use the Cauchy Condensation Test.
1. Rewrite the following as a single logarithm: log p- (log 9-3 log y (2 marks) 2 2. Given a3x = 2 and a®) = 9, then determine an expression that represents the value of log, 18 . (2 marks) 3. A student's work to simplify a logarithmic expression as shown below, where a >1 Step 1 Step 2 2 log, ** – 3 log, x? +4log, x' log, ** – log, x° + log, x'? x Step 3 log....
is this correct Write the expression as a single logarithm. 1 2 loga (6x) - log (2x+15) 14 36x 2 loga (6x?) - log a(2x+15)= loga 1 9 (2x + 15) (Simplify your answer.)
Rewrite the expressions as a single logarithm. log. x-3 -2 log.y-3log. x+1
R R 5. To compute 1 = lim 2 COS dr and J = lim 22+1 sinc dx simultaneously .22 +1 R R0 R R using Residue Theorem, let f(x) 22 +1 C COSC sinc (1) Show that if z = x + iy, then Rf(R2) = and Sf(R2) = x2 +1 x2 +1 (2) Find Res[f, i]. (3) Show that I = 0 and J (4) Prove I = 0) in the above problem without using Residue Theorem. IT
10. Let R > 0, Describe the arc「with parametrisation z(t)-Reit for-π/2-t-π/2 Use th is parametrisation to calculate the integral log z dz, where log z denotes the principal value of the logarithm.