Determine if the set {sin^2(x),1-cos(2x)} is linearly dependent or linearly independent on I = (−∞,∞)

Consider the complex function f(z)= 1/(z^2 +1)  Find the residue of f(z) at the pole z=i.

Write the general solution ot cn ODE which has the characteristic equation... (^-4)² (1+7) (^'^*^+1) (1`°+9)) = 0 о

When a line 'I' passes through the origin (0,0), it divides the line 'l' into two distinct rays. Furthermore, when considering a complex-valued function 'g' defined on a set 'S,' we say that 'g' is bounded if there exists a finite constant 'M' such that the absolute value of 'g' (Igl) is always less than 'M' for all points in 'S. Now, we aim to determine for which of these rays passing through the origin the function e^z satisfies this boundedness property Write the following in a + bi form. e4ln 3+i Log(4 + 4i) Log(-37)