Cauchy-Riemann equations, vector gradients and contour lines

1. a) The Cauchy-Riemann equation is the name given to the following pair of equations, ∂u/∂x=∂v/∂y and ∂u/∂y= -∂v/∂x which connects the partial derivatives of two functions u(x, y) and v(x, y) i) if u(x, y) =e^x cosy and v(x, y) =e^x siny, how do I prove that these functions ...continues

Functions And Notation

1. From the function f (x)=IxI How would I go about finding its image set using interval notation? 2. Again using interval notation, how would I go about finding the image set of the graph g(x)= Ix+3I -2 ? And how then would I go on to solve the equation g(x)=1, and discover if it had any geometrical significance? 3. How ...continues

Domain and image sets

If I have the function f(x)=2e x+1 , how would I discover the domain and image set of function f? How would I go about finding the domain and image set of f -1? How would I go about solving the equation y= 2e x+1, to find x in terms of y?

L'Hospital's Rule, Asymptotes, Global Extrema, Inflection Points and Concavity : f(x) = x^2 e^(17x)

f(x) = x^2 e^(17x) 1. Find an equation for each horizontal asymptote to the graph of f. 2. Find an equation for each vertical asymptote to the graph of f. 3. Determine all critical numbers. 4. Determine the global maximum of the function. 5. Determine the global minimum of the function. 6. Find inflection points. 7. Fin ...continues

Contrast and Discuss Asymptotes and a Real-Life Example of an Exponential Function

One of the archeologists you interviewed for your article is graphing asymptotes to illustrate the data generated through carbon dating the half-life of fossil specimens. Help him with his work by solving these problems: 1. Explain and contrast the types of asymptotes considered for ratio ...continues

Compactness with two equivalent norms

(See attached file for full problem description and symbols) --- Assume that and are two equivalent norms on X, and that . Prove that M is compact in if and only if M is compact in .

Maximum and Minimum theorem

Prove: A continuous mapping T of a compact subset M of a metric space X into assumes a maximum and a minimum at some points of M.

Functional Analysis

Attachment file. Let X be a normed space and . Show that if for every bounded linear functional f on X , then .

Limit Inferior

Suppose a_n >0 for each n in N and lim inf (a_n) > 0. Prove there is a number a>0 st a_n >/= a for all n in N. (limit n--> infinity)

Bounded Proof : If {a_n + b_n} and {a_n} are both bounded, then {b_n} is bounded.

If {a_n + b_n} and {a_n} are both bounded, then {b_n} is bounded.