Answer to Question #93684 in Complex Analysis for Edward

Question #93684
U(x,y)+iv(x,y) a. Z^2+2iz b. e^z2 c. In(1+z)
1
Expert's answer
2019-09-06T09:52:24-0400

a)

"z^2=(x+iy)^2=x^2-y^2+2ixy"

"2iz=2i(x+iy)=-2y+2ix"

"z^2+2iz=x^2-y^2-2y+2ixy+2ix-(x^2-y^2-2y)+i(2x+2xy)"

"U=x^2-y^2-2y;\\quad V=2x+2xy"

b)

"e^{z^2}=e^{(x+iy)^2}=e^{x^2-y^2+2ixy}=e^{x^2-y^2}e^{2ixy}=e^{x^2-y^2}(\\cos(2xy)+i\\sin(2xy))"

"U=e^{x^2-y^2}\\cos(2xy);\\quad V=e^{x^2-y^2}\\sin(2xy)"

c)

"ln(1+z)=ln(1+x+iy)"

"lnz=ln|z|+i(\\arg(z)+2\\pi k)"

"|z+1|=\\sqrt{(x+1)^2+y^2}"

"\\arg(z+1)=\\arctan\\frac{y}{x+1}"

"ln(z+1)=ln\\sqrt{(x+1)^2+y^2}+i(\\arctan\\frac{y}{x+1}+2\\pi k)"

"U=ln\\sqrt{(x+1)^2+y^2} ;\\quad V=\\arctan\\frac{y}{x+1}+2\\pi k"


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