Example 7. Calculate the integral

$$
I=\int_{1}^{i} \frac{\ln ^{3} z}{z} d z
$$

along the arc of the circle $|z|=1$ (where $\ln z$ is the principal value of the logarithm, $\ln 1=0$).
Solution. First method. Applying the Newton-Leibniz formula, we get

$$
I=\int_{1}^{i} \frac{\ln ^{3} z}{z} d z=\int_{1}^{i} \ln ^{3} z d(\ln z)=\left.\frac{\ln ^{4} z}{4}\right|_{1} ^{1}=\frac{\ln ^{4} i-\ln ^{4} 1}{4}=\frac{\ln ^{4} i}{4}=\frac{1}{4}\left(\frac{\pi i}{2}\right)^{4}=\frac{\pi^{4}}{64}
$$

Second method. We make a change of variable

$$
\ln z=w, \quad d w=\frac{d z}{z}
$$

The arc of the circle $|z|=1$ transforms into a segment of the imaginary axis, enclosed between the points $(0,0)$ and $\left(0, \frac{\pi}{2}\right)$.

The integral will take the form

$$
I=\int_{0}^{i \pi / 2} w^{3} d w=\left.\frac{w^{4}}{4}\right|_{0} ^{i \pi / 2}=\frac{1}{4} \frac{\pi^{4} i^{4}}{2^{4}}=\frac{\pi^{4}}{64}
$$

Third method. Let $z=e^{i \varphi}$ (here $\rho=|z|=1$). Then

$$
\ln z=i \varphi, \quad d z=i e^{i \varphi} d \varphi
$$

The real variable $\varphi$ varies within the limits $0 \leqslant \varphi \leqslant \frac{\pi}{2}$: In this case, we get

$$
I=\int_{0}^{\pi / 2} \frac{i^{3} \varphi^{3} e^{i \varphi} i d \varphi}{e^{i \varphi}}=\int_{0}^{\pi / 2} \varphi^{3} d \varphi=\left.\frac{\varphi^{4}}{4}\right|_{0} ^{\pi / 2}=\frac{\pi^{4}}{64}
$$

## Problems for Independent Solution

Calculate the following integrals:

140. $\int_{C} z \operatorname{Im} z^{2} d z, C:|z|=1(-\pi \leqslant \arg z \leqslant 0)$.
141. $\int_{C} e^{|z|^{2}} \operatorname{Re} z d z, C-$ a straight line connecting the points $z_{1}=0, z_{2}=1+i$.
142. $\int_{C} \ln z d z(\ln z-$ principal value of the logarithm), $C:|z|=1$, a) initial point of the integration path $z_{0}=1 ;$ b) $z_{0}=-1$. Counterclockwise.
143. $\int_{C} z \operatorname{Re} z d z, C:|z|=1$. Counterclockwise.
144. $\int_{C} z \bar{z} d z ; C:|z|=1$ Counterclockwise. 145. $\int_{1}^{1} z e^{z} d z$.
145. $\int_{C} \operatorname{Re} z d z, C:$ a) $z=(2+i) t(0 \leqslant t \leqslant 1)$; b) a broken line consisting of the segment $[0,2]$ of the real axis and the segment connecting the points $z_{1}=2 z_{2}=2+i$.
146. $\int_{1+i}^{-1-i}(2 z+1) d z$.
147. $\int_{0}^{i+1} z^{3} d z$
148. $\int_{1}^{1}\left(3 z^{4}-2 z^{3}\right) d z$.
149. $\int_{C} e^{z} d z, C$ a) arc of the parabola $y=x^{2}$, connecting the points $z_{1}=0$ and $z_{2}=1+$; b) segment of the straight line connecting these same points.
150. $\int_{C} \cos z d z, C$ : segment of the straight line connecting the points $z_{1}=\frac{\pi}{2}$ and $z_{2}=\pi+i$. 152. $\int_{C} \frac{d z}{\sqrt{2}}, C$ : a) upper half of the circle $|z|=1$; the branch of the function $\sqrt{2}$ is chosen such that $\sqrt{1}=1 ;$ b) $|z|=1, \operatorname{Re} z \geqslant 0 ; \sqrt{-i}=(1-i) \frac{\sqrt{2}}{2}$.
151. $\int_{C} \frac{d z}{\sqrt[4]{z^{3}}}, C$ : upper half of the circle $|z|=1$; the branch of the function $\omega=\sqrt[4]{z^{3}}$ is chosen such that $\sqrt[4]{1}=1$.
152. $\int_{1+i}^{2 i}\left(z^{3}-z\right) e^{z^{2} / 2} d z$. 155. $\int_{0}^{i} z \cos z d z$.
153. $\int_{1}^{i} z \sin z d z$
154. $\int_{0}^{i}(z-i) e^{-z} d z$.
155. $\int_{1}^{i} \frac{\ln (z+1)}{z+1} d z$ along the arc of the circle $|z|=1, \operatorname{Im} z \geqslant 0, \operatorname{Re} z \geqslant 0$.
156. $\int_{1}^{i} \frac{\ln z}{z} d z$ along the segment of the straight line connecting the points $z_{1}=1$ and $z_{2}=i$.
157. $\int_{0}^{1+i} \sin z \cos z d z$
158. $\int_{1}^{i} \frac{1+\operatorname{tg} z}{\cos ^{2} z} d z$ along the straight line connecting the points $z_{1}=1$ and $z_{2}=i$.
159. $\int_{-1}^{i} \frac{\cos z d z}{\sqrt{\sin z}}$ along the straight line connecting the points $z_{1}=-1$ and $z_{2}=i$; the branch of the function $\sqrt{\sin z}$ is chosen such that $\sqrt{\sin (-1)}=i \sqrt{\sin 1}$.
160. $\int_{C} \operatorname{Re}(\sin z) \cos z d z, C:|\operatorname{Im} z| \leqslant 1, \operatorname{Re} z=\frac{\pi}{4}$.
161. $\int_{C} z \operatorname{Im}\left(z^{2}\right) d z, C:|\operatorname{Im} z| \leqslant 1, \operatorname{Re} z=1$. 165. $\int_{-i}^{i} z e^{z^{2}} d z$
162. $\int_{C} \operatorname{tg} z d z, C:$ arc of the parabola $y=x^{2}$, connecting the points $z=0, z=1+i$.

## § 6. Cauchy's Integral Formula

If the function $f(z)$ is analytic in a domain $D$ bounded by a piecewise-smooth closed contour $C$, and on the contour itself, then the Cauchy integral formula holds

$$
f\left(z_{0}\right)=\frac{1}{2 \pi i} \int_{C} \frac{f(z) d z}{z-z_{0}} \quad\left(z_{0} \in D\right)
$$

where the contour $C$ is traversed such that the domain $D$ remains on the left side.

The Cauchy integral formula allows the calculation of certain integrals.