---
title: "Analyticity of a function"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:analyticity-of-a-function"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/analyticity-of-a-function
site: BCT Engineering
---

# Analyticity of a function

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2081 Chaitra (new course), 2 marks
- Video: [Complex Analysis | Analytic Function | Cauchy Riemann Equation BY GP sir](https://www.youtube.com/watch?v=t9xW7UaZwZ0) — Dr.Gajendra Purohit and 2 more
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/analyticity-of-a-function
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

For a complex variable, show that f(z) = sin z is analytic.

## Short answer

Writing sin z = sin x cosh y + i cos x sinh y, the partial derivatives are continuous and satisfy u_x=v_y, u_y=-v_x everywhere. Hence sin z is entire with derivative cos z.

## Answer

A function f(z) = u + iv is analytic in a region if u and v have continuous first partial derivatives there and satisfy the Cauchy–Riemann equations u_x = v_y and u_y = −v_x.

Write z = x + iy and use sin(x + iy) = sin x cos(iy) + cos x sin(iy) = sin x cosh y + i cos x sinh y.

```text
u = sin x cosh y,    v = cos x sinh y

u_x = cos x cosh y    v_y = cos x cosh y   → u_x = v_y ✓
u_y = sin x sinh y    v_x = −sin x sinh y  → u_y = −v_x ✓
```

The partial derivatives are continuous for all x, y, and the Cauchy–Riemann equations hold everywhere. Hence **sin z is analytic everywhere** (it is an entire function), and f'(z) = u_x + i v_x = cos x cosh y − i sin x sinh y = cos z.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. f = u+iv is analytic if:
   A. u = v
   B. u_x = v_x
   C. u,v have continuous first partials satisfying Cauchy-Riemann
   D. only u is continuous
   Answer: C. u_x = v_y, u_y = -v_x. (Key point: Condition)

2. sin(x+iy) =
   A. sin x cos y + i cos x sin y
   B. sin x sinh y + i cos x cosh y
   C. cos x cosh y + i sin x sinh y
   D. sin x cosh y + i cos x sinh y
   Answer: D. Using cos(iy)=cosh y and sin(iy)=i sinh y. (Key point: Split)

3. u_x = v_y because both equal:
   A. cos x cosh y
   B. sin x sinh y
   C. cos x sinh y
   D. -sin x cosh y
   Answer: A. u_x = cos x cosh y, v_y = cos x cosh y. (Key point: CR check)

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