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freckles
 one year ago
\[\text{ Suppose } f \text{ is an analytic function of the complex variable } z=x+i y \text{ given by } \\ f(z)=(2x+3y)+i g(x,y) \\ \text{ where} g(x,y) \text{ is a realvalued function } \\ \text{ of the real variables } x \text{ and } y . \\ \text{ If } g(2,3)=1\text{, then }g(7,3)=? . \]
freckles
 one year ago
\[\text{ Suppose } f \text{ is an analytic function of the complex variable } z=x+i y \text{ given by } \\ f(z)=(2x+3y)+i g(x,y) \\ \text{ where} g(x,y) \text{ is a realvalued function } \\ \text{ of the real variables } x \text{ and } y . \\ \text{ If } g(2,3)=1\text{, then }g(7,3)=? . \]

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freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[f(2+3i)=4+9+ig(2,3) \\ f(2+3i)=13+i g(2,3) \\ f(2+3i)=13+i \\ f(7+3i)=14+9+ig(7,3) \\ f(7+3i)=23+i g(7,3) \\ \] this as far as I can get so far

freckles
 one year ago
Best ResponseYou've already chosen the best response.1There are choices A) 14 B) 9 C) 0 D) 11 E) 18 This question comes from the mathematics gre practice exam.

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0hmm interesting, i have some admiration for complex though i have no skills in such things yet lol

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I believe it might have something to do with cancelling the complex part.

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0i have a feeling that it is not that hard just kind of a trick

freckles
 one year ago
Best ResponseYou've already chosen the best response.1was kind of thinking that too

freckles
 one year ago
Best ResponseYou've already chosen the best response.1somehow we have to show it is 14

freckles
 one year ago
Best ResponseYou've already chosen the best response.1maybe and I could be totally crazy we could do something with magnitudes of complex numbers

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[g(2,3)=1 \\ f(2,3)=\sqrt{170} \\ g(7,3)=\sqrt{a^2} \text{ if } g(7,3)=a \\ f(7,3)=\sqrt{23^2+g^2(7,3)}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[f^2(7,3)=23^2+a^2 \]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0you 14 was the answer! did you just tried it directly?

freckles
 one year ago
Best ResponseYou've already chosen the best response.1no I looked at the answers

freckles
 one year ago
Best ResponseYou've already chosen the best response.1how did you know 14 was right?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0no i just saw your reply above mentioning that

freckles
 one year ago
Best ResponseYou've already chosen the best response.1I wonder what analytic means

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0isn't that related to graphs ?

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0don't know why wolfram ignores the imaginary stuff for such functions

freckles
 one year ago
Best ResponseYou've already chosen the best response.1https://www3.nd.edu/~atassi/Teaching/ame60612/Notes/analytic_functions.pdf go to page 2

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[f(x,y)=u(x,y)+i v(x,y) \\ \frac{\partial u}{ \partial x}=\frac{\partial v}{ \partial y} \\ \frac{\partial u}{ \partial y}=\frac{ \partial v }{\partial x} \\ 2=g_y \\ 3=g_x\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Analytic functions refers to something else. One important property of analytic functions are that they are indefinitely differentiable.

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0you know it crossed my mind that we need to use calculus lol

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[g(x,y)=2y+C(x) \\ g(x,y)=3x+K(y) \\ \\ 0=2y+3x+C(x)+K(y) \] where C(x) is a function of x and K(y) is a function of y

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0so any analytic function has no differentiability problem is that what you are saying?

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[2g(x,y)=2y3x+C(x)K(y) \\ 2g(2,3)=2(3)3(2)+C(2)K(3) \\ g(2,3)=C(2)K(3)\]

xapproachesinfinity
 one year ago
Best ResponseYou've already chosen the best response.0oh i see! after checking the theorem in that pdf

freckles
 one year ago
Best ResponseYou've already chosen the best response.1lol still trying to play with this fungus

freckles
 one year ago
Best ResponseYou've already chosen the best response.1and that is equal to 1

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[2g(7,3)=2(3)+3(7)+C(7)K(3)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[2g(7,3)=27+C(7)K(3)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[2g(7,3)1=27+C(7)C(2)\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Oh yea, use the Cauchy–Riemann equations to solve this.

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[2g(7,3)=28+C(7)C(2) \\ g(7,3)=14+\frac{C(7)C(2)}{2}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1oops i made a typeo somewhere above

freckles
 one year ago
Best ResponseYou've already chosen the best response.1\[g(x,y)=2y+C(x) \\ g(x,y)=3x+K(y) \\ \\ 0=2y+3x+C(x)+K(y)\] \[2g(x,y)=2y3x+C(x)K(y) \\ 2g(2,3)=2(3)3(2)+C(2)K(3) \\ g(2,3)=C(2)K(3)\] \[C(2)K(3)=1 \\ 2g(7,3)=621+C(7)K(3) \\ 2g(7,3)=15+C(7)K(3) \\ 2g(7,3)1=15+C(7)K(3)C(2)+K(3) \\ 2g(7,3)=14+C(7)C(2) \\ g(7,3)=7+\frac{C(7)C(2)}{2}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.1I think I'm still making a mistake somewhere

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You simply use the CauchyRiemann equations above, and calculate the partials. There, you had \[2 = g_{y}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0The CR equations are your best bet. It becomes a problem similar to finding potential functions.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yes, and then you find the potential function by adding the constant term.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So your potential function g(x,y) becomes 3x + 2y + C

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0But you are given an initial condition, g(2,3)=1. Plugging it back into our potential function, you will find that C = 1.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Therefore, you get \[g(x,y) = 3x+2y+1\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So just plug in, g(7,3) and you will get 14.

freckles
 one year ago
Best ResponseYou've already chosen the best response.1@Loser66 check this out

freckles
 one year ago
Best ResponseYou've already chosen the best response.1I will still need to look at that theorem thing because I think I was using a little wrong. lol like I thought I was supose to write g_x=3 implies g(x,y)=3x+K(y) where K is a function of y and g_y=2 implies g(x,y)=2y+C(x) where C is a function of x but I kept getting ugly stuff

freckles
 one year ago
Best ResponseYou've already chosen the best response.1thanks for your help guys
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