Complex Analysis (MP233CC1) - Quiz 1

If $$f\left(t\right)=u\left(t\right)+iv\left(t\right)$$ is a continuous function defined in an interval $$\left(a,\ b\right),\$$ then the value of the integral $$f\left(t\right)$$ is defined to be $$\int_a^bf\left(t\right)dt\ =\int_a^bu\left(t\right)dt+i\int_a^bv\left(t\right)dt$$

If $$f\left(t\right)=u\left(t\right)+iv\left(t\right)$$ is a continuous function defined in an interval $$\left(a,\ b\right),\$$ then the value of the integral $$f\left(t\right)$$ is defined to be $$\int_a^bf\left(t\right)dt\ =\int_a^b\left(u\left(t\right)dt+iv\left(t\right)\right)dt$$

If $$\gamma$$ is a piecewise differentiable arc with $$z=z\left(t\right),\ a\le t\le b$$ and $$f\left(z\right)$$ is continuous on $$\gamma,$$ then $$\int_{\gamma}^{ }f\left(z\right)dz=\int_a^bf\left(z\left(t\right)\right)z'\left(t\right)dt$$

If $$\gamma$$ is a piecewise differentiable arc with $$z=z\left(t\right),\ a\le t\le b$$ and $$f\left(z\right)$$ is continuous on $$\gamma,$$ then $$\int_{\gamma}^{ }f\left(z\right)dz=\int_a^bf\left(u\left(t\right)\right)u'\left(t\right)dt+i\int_a^bf\left(v\left(t\right)\right)v'\left(t\right)dt$$

The area enclosed by both curves will be zero.

True

False

True

False

1

0

Winding number of a with respect to $$\gamma$$

Representation Formula

Representation of $$\gamma$$

$$\frac{M}{r}$$

$$\frac{M}{r^n}$$

$$\frac{n!M}{r}$$

$$\frac{n!M}{r^n}$$

True

False