anonymous
  • anonymous
Test the analyticity of the function w=sin z hence show that the derivative of sin z = cos z.
Differential Equations
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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experimentX
  • experimentX
to show function is analytic , you have to show that it satisfies CR equations and have continuous partial derivatives. derivative of sin(z) is cos(z) follows from following definition. \[ \lim_{\Delta z \to 0}\frac{\sin(z+\Delta z) - \sin(z)}{\Delta z} = \cos z\]
anonymous
  • anonymous
What of analyticity? If I could just get a clear working,I can be able to tackle other problems.
experimentX
  • experimentX
sin(z) = sin(x+iy) <-- can you show that it satisfies CR equations? just like i did in your other question

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experimentX
  • experimentX
sin(z) = (e^(iz) - e^(iz))/(2i) the simplest logic would be to use the definition like above. you know exponential function is entire function and the sum of entire function is also entire. so sin(z) is also an entire function i.e. holomorphic in entire domain. holomorphic function are analytic.
anonymous
  • anonymous
Thanks but we are not deep into analytic functions yet,I've tried the other but got stuck midway,please show me a clear working of the problem.
experimentX
  • experimentX
show me your working first ... it will help me to help you.
anonymous
  • anonymous
hey..there's a working we have tried but got stuck while testing the partial derivatives..we have a problem keying in our working for both problems.
experimentX
  • experimentX
if you have typed, it would make thins easier for me .. anyway let me write it in short.
anonymous
  • anonymous
we would appreciate.
experimentX
  • experimentX
\[ \sin(x+iy)= \sin(x)\cos(iy) + \sin(iy)\cos(x) = \sin(x)\cosh(y) + i \cos(x) \sinh(y) \\ = u(x,y) + v(x,y)\] ^^ test the above function for CR equations. I have used following formula \[ \sin(ix) = i \sinh(x) \\ \cos(ix) = \cosh(x)\]
experimentX
  • experimentX
also it is not difficult to show that they are continuous on entire domain. because they are just product of two continuous function. to find the derivative, just use the usual definition used in elementary calculus.
anonymous
  • anonymous
Thank you.
anonymous
  • anonymous
thank you ..i appreciate your help
experimentX
  • experimentX
you are welcome ... also to find the derivative, you can directly do \[ f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y}\] which you should get u+iv form of cos(x+iy)
anonymous
  • anonymous
Close, @experimentX, but the Wirtinger derivative has a \(1/2\). https://en.wikipedia.org/wiki/Wirtinger_derivative
experimentX
  • experimentX
that comes from chain rule ... dw/dx = dw/dz.dz/dx + dw/dzbar. dzbar/dx do same for y part and solve for dw/dz, you should get 1/2

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