九州大学 システム情報科学府 情報理工学専攻・電気電子工学専攻 2024年8月実施 解析学・微積分

Author

祭音Myyura (assisted by ChatGPT 5.5 Thinking)

Description

(1) Calculate the following integral, where and are positive constants and denotes the set of all real numbers.

(2) Find the solution to the following differential equation using Laplace transforms. Here, denotes the first-order derivative of a function with respect to .

(3) Let , where and are real numbers, and . Answer the following questions.

• (a) Find a real number for which the function is holomorphic.
• (b) Consider a holomorphic function . Suppose the real part of the function is given as where denotes the real part of the function . Then, find a formula for the function .

Kai

(1)

We calculate

where

Use the change of variables

where

The Jacobian is

Also,

Therefore,

Since

and

we get

Hence,

(2)

Let

Taking the Laplace transform of both sides of

we have

Using

and the initial conditions

we obtain

Since

we have

Thus,

Hence,

Now decompose

Then

Comparing coefficients gives

Therefore,

Thus,

Taking the inverse Laplace transform, we get

Therefore,

Hence,

(3)

(a)

Let

Write

where

For to be holomorphic, the Cauchy-Riemann equations must hold:

Compute each derivative:

The first Cauchy-Riemann equation gives

Thus,

The second Cauchy-Riemann equation also gives the same condition:

Hence,

Indeed, when ,

which is holomorphic.

(b)

We are given

Recall that

Therefore, the real part of is

Hence, one such holomorphic function is

Since adding a purely imaginary constant does not change the real part, the general formula is

where is a real constant.