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Lösung 3.1:4f

Aus Online Mathematik Brückenkurs 2

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K (Lösning 3.1:4f moved to Solution 3.1:4f: Robot: moved page)
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<center> [[Image:3_1_4f-1(2).gif]] </center>
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Because the equation contains both <math>z</math> and <math>\bar{z}</math>, we cannot use <math>z</math> (or <math>\bar{z}</math>) alone as an unknown, so we are forced to set
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<center> [[Image:3_1_4f-2(2).gif]] </center>
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<math>z=x+iy</math>
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and use the real part <math>x</math> and the imaginary part <math>y</math> as unknowns.
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With this approach, the left-hand side of the equation becomes
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<math>(1+i)(x-iy)+i(x+iy)
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\begin{align} &=1\cdot x -1\cdot iy +i\cdot x -i^2y + i\cdot x + i^2y\\
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&=x-iy+ix+y+ix-y\\
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&=x+(2x-y)i\end{align}</math>
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and the whole equation becomes
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<math>x+(2x-y)i=3+5i</math>.
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The two complex numbers on the left- and right-hand sides are equal when both real and imaginary parts are equal, i.e.
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<math>\begin{cases}x=3\\2x-y=5\end{cases}</math>.
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This gives <math>x=3</math> and <math>y=2x-5=2\cdot 3-5=1</math>. Thus, the equation has the solution <math>z=3+i</math>.
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A quick check shows that <math>z=3+i</math> satisfies the equation in the exercise:
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<math>\begin{align}LHS &= (1+i)\bar{z}+iz=(1+i)(3-i)+i(3+i)\\
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&= 3-i+3i+1+3i-1 = 3+5i = RHS\end{align}</math>
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Version vom 11:45, 23. Sep. 2008

Because the equation contains both z and z, we cannot use z (or z) alone as an unknown, so we are forced to set


z=x+iy


and use the real part x and the imaginary part y as unknowns.

With this approach, the left-hand side of the equation becomes


(1+i)(x−iy)+i(x+iy)=1x−1iy+ix−i2y+ix+i2y=x−iy+ix+y+ix−y=x+(2x−y)i


and the whole equation becomes


x+(2x−y)i=3+5i.


The two complex numbers on the left- and right-hand sides are equal when both real and imaginary parts are equal, i.e.


x=32x−y=5 .


This gives x=3 and y=2x−5=23−5=1. Thus, the equation has the solution z=3+i.

A quick check shows that z=3+i satisfies the equation in the exercise:


LHS=(1+i)z+iz=(1+i)(3−i)+i(3+i)=3−i+3i+1+3i−1=3+5i=RHS

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.1:4f“