Question

Find the value(s) of constant B such that :

integral on curve for [ (1)- (3z) + (2 B {z^4}) +( z^6) + (3 {z^7}) + ( 11 {z ^ 100} ] /[ z^5] dz =
integral on curve for [ e ^ {Bz} + 2 z ] / [ z^3 ] dz
where C is the unit circle oriented counterclockwise

Answer 1

Answer on Question #50235 – Math - Complex Analysis

Find the value(s) of constant B such that:

integral on curve for $[(1)-(3z)+(2B\{z^4\})+(\z^6)+(3\{z^7\})+(\11\{z^{\wedge}100\}] / [\z^5]\ \mathrm{d}z =$ integral on curve for $[e^{\wedge}\{Bz\} + 2z] / [z^3]\ \mathrm{d}z$

where $C$ is the unit circle oriented counterclockwise.

Solution

$$I_1 = \oint f_1(z) \, dz = I_2 = \oint f_2(z) \, dz$$

where $f_1(z) = \frac{1 - 3z + 2Bz^4 + z^6 + 3z^7 + 11z^{100}}{z^5}$, $f_2(z) = \frac{e^{Bz} + 2z}{z^3}$.

Both integrands have singularities at $z = 0$ and the value of each integral can be evaluated as $I_1 = 2\pi \iota \cdot res[f_1(z), z = 0] = I_2 = 2\pi \iota \cdot res[f_2(z), z = 0]$.

We can break up the integrand $f_1(z)$ into six fractions dividing each term in the numerator by the denominator $f_1(z) = \frac{1 - 3z + 2Bz^4 + z^6 + 3z^7 + 11z^{100}}{z^5} = \frac{1}{z^5} - 3\frac{1}{z^4} + 2B\frac{1}{z} + z + 3z^2 + 11z^{95}$, we see that $f_1(z)$ has a Laurent expansion about $z = 0$ given by

$f_1(z) = \frac{1}{z^5} - 3\frac{1}{z^4} + 2B\frac{1}{z} + z + 3z^2 + 11z^{95}$. Hence the residue is $2B$ (the coefficient of $z^{-1}$).

$$\oint \frac{1 - 3z + 2Bz^4 + z^6 + 3z^7 + 11z^{100}}{z^5} \, dz = 2\pi \iota \cdot res[f_1(z), z = 0] = 2\pi \iota c_{-1} = 2\pi \iota 2B$$

$c_{-1}$ is the coefficient of the Laurent expansion about $z = 0$.

In a similar way by expanding $\frac{e^{Bz} + 2z}{z^3}$ as a Taylor series, we see that

$f_2(z) = \frac{e^{Bz} + 2z}{z^3} = \frac{1 + Bz + B^2z^2 / 2 + B^3z^3 / 6 + 2z}{z^3}$ has a Laurent expansion about $z = 0$ given by $\frac{1}{z^3} + \frac{B + 2}{z^2} + \frac{B^2}{2z} + B^3 + \ldots$

So, the residue is $\frac{B^2}{2}$ (the coefficient of $z^{-1}$). Hence the both integrals are equal if the next condition is fulfilled: $B^2 / 2 = 2B \Rightarrow \begin{cases} B = 0 \\ B = 4 \end{cases}$.

Answer: $\begin{cases} B = 0 \\ B = 4 \end{cases}$

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