(a) Determine which of the following functions are analytic....

(a) Determine which of the following functions are analytic. If Analytic find its derivative. (i) (x+iy)/(x^2+y^2) (ii) cosh z (b) Prove that u = x^2 - y^2 - 2xy - 2x + 3y is harmonic .Find a function v such that f(z) = u + iv is analytic. Also express f(z) in terms of z.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Transcript

00:01 Okay, so let's get started with the first exercise.

00:05 Well, for part i of our exercise, we just need to observe that x plus iy over x squared plus y squared is equal to z over the length of z squared.

00:20 And this one is not holomorphic.

00:26 It is not holomorphic at z equal to zero.

00:31 Now, the second part.

00:33 Well, this one is holomorphic.

00:37 Hyperbolic cosine of z is e to the z plus e to the negative z over 2.

00:45 Now, what is the derivative? well, the derivative is hyperbolic sine.

00:58 Okay, perfect.

01:02 This one is negative hyperbolic sine.

01:08 Okay, now let's go on.

01:11 Part b of our exercise.

01:14 Well, here we need to prove that u equal to x squared minus y squared minus 2xy minus 2x plus 3y is harmonic.

01:28 How can we do this? well, we want a function, a holomorphic function f of z such that f of z is equal to u plus iv.

01:40 Okay, now this function is holomorphic if and only if it satisfies the cauchy -riemann equations.

01:48 That is, the partial derivative of u with respect to x must be equal to the partial derivative of v with respect to y.

01:57 And the partial derivative of u with respect to y must be equal to negative 1 multiplied by the partial derivative of v with respect to x.

02:10 Okay, so let's get started with the first equation...

Question

(a) Determine which of the following functions are analytic. If Analytic find its derivative. (i) (x+iy)/(x^2+y^2) (ii) cosh z (b) Prove that u = x^2 - y^2 - 2xy - 2x + 3y is harmonic .Find a function v such that f(z) = u + iv is analytic. Also express f(z) in terms of z.

Please give Ace some feedback