Complex Proofs2017-02-11T11:51:35+10:00

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Complex Proofs

$$z\in C$$ such that $$\frac{z}{z-i}$$ is real. Show that z is imaginary.

Let $$\frac{z}{z-i}=a$$, where a is real.

Then $$z=a(z-i)$$

$$z=az-ai$$
$$ai=az-z$$
$$ai=z(a-1)$$

so $$z=\frac{ai}{a-1}$$

$$z=\frac{a}{a-1}i$$

Since a is real, then $$\frac{a}{a-1}$$ is real, and hence $$\frac{a}{a-1}i$$ is imaginary

thus z is imaginary

 

 

further examples

= 10x + 15 – 7x + 14 = 3x + 29

harder examples

u = 1 + √x = 1 + x1/2 du/dx = ½x–1/2 du = $$\frac{1}{{2\sqrt x }}dx$$ x = 49 u = 8 x = 1 u = 2 = $$2\int\limits_2^8 {\frac{1}{{2\sqrt x \sqrt u }}dx} $$ = $$2\int\limits_2^8 {{u^{ – 1/2}}\,du} $$ = $$2\left[ {\frac{{{u^{1/2}}}}{{1/2}}} \right]_2^8$$ = $$4\left[ {\sqrt u } \right]_2^8$$ = 4{√8 – √2} = 4(2√2 – √2) = 4 × √2 = 4√2 = √32

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